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0, u, v E V t , and u > v, then 0 must blow up in finite time, by Proposition 2.1 and because 0 is unstable. Therefore A+ can be considered also as a threshold for blow-up of all positive orbits of <po This point of view was taken by Il'yasov,124-126 who proves the following characterization of A+: 0 in n for all iE{l, ... ,n}; (ii) (N5) holds and there is 0 in n for some iE{l, ... ,n}. Then for each holds for all X E 0 and i E S as soon as it holds for some X E 0 and i E S. Hence either in O. Without any strict positivity assumption, Theorem 5.1 is not true even in the scalar case (see Section 3.1), but we have the following partial symmetry result similar to Theorem 3.2. 0, and lim sup
12
smooth functions the result follows. 9 Alternatively, one uses directly in the variations-of-constants formula that - ~ is resolvent positive. 84 Strong monotonicity has striking consequences for the qualitative behavior of the flow. It is used, for example, in the study of convergence of relatively compact semiorbits, see the articles by Hirsch 85 and Polacik. 86 It structures state space and leads to results on existence and qualitative aspects of equilibria and connecting orbits. A much richer structure exists for equations in one space dimension, the zero number or lap number. 87- 91 Assume that n is a bounded open interval in lEt For a continuous function u: 0 ----+ lR that does not vanish everywhere define its zero number (( u) to be the supremum of all n E No such that there is a sequence of points Xo < Xl < ... < Xn in 0 with U(Xk-I)U(Xk) < 0 for k = 1, 2, ... n. Then (: E\ {O} ----+ No U {oo} is well defined because E <:;; C(O). As long as u,v E V t , ((cpt(u) - cpt(v)) is a decreasing function of t. Similarly, if vEE and as long as U E Vi> ((Dcpt(u)v) is decreasing in
t. The zero number gives even better results in the qualitative study of the parabolic flow. In contrast to the higher dimensional case, here a relatively compact positive semiorbit always converges to its w-limit set. 92 ,93 A result by Henry 94 states that if the equilibria are all nondegenerate then the stable and unstable manifolds intersect transversally along a connecting orbit. If one considers problems with a global attract or (e.g., the ChafeeInfante problem) then using the zero number one can show that the attractor is a smooth graph over a finite dimensional subspace of E. 95 ,96 Moreover, it allows to characterize precisely what pairs of equilibria have a connecting orbit. 97 ,98 A similar result has recently been shown in the definite superlinear case by Schulz. 99 The blow-up dynamics in the strong unstable manifold of 0 is analyzed by Fiedler and Matano. IOO Finally, note that in the higher dimensional case there does not seem to exist a structure that is equivalent to the zero number. lOl
2.4. Blow- Up in Finite Time and A Priori Bounds Solutions of (P.x) that blow up in finite time and bounds for globally existing solutions have been studied intensely since the 1980s, see for example the survey [10J on blow-up. One of the most comprehensive sources for results on blow-up and a priori bounds in superlinear problems is the book by Quittner and Souplet. 68 Since we are only concerned with the long time dynamics, i.e., equilibria and connecting orbits of (P .x), we only present what is useful here and do not go into describing the history of the development
13
of a priori bounds. First we define the constant PR that appears in the statement of condition (I) (it is the constant from [32]):
pz
PR :=
00
N= 1,
5
N=2,
3
N=3,
8
3
N=4,
9N 2 -4N+16!..N(N -1) (3N _4)2
N::::5.
(5)
The following are the consequences of the results in [16,32] that we will need: Proposition 2.1. Every global positive semiorbit is bounded and hence relatively compact. Let M be a closed bounded subset of E. Then A(M) <;;; g. Moreover, if A <;;; A(M) is relatively compact then the positive semiorbit O+(A) is relatively compact. Proof. The first assertion is proved in [16, Theorem 1.1] and [32, Theorem 3.13]. If u E A(M) then O+(u) remains bounded, i.e., u E g. If u E A(M) and t E [0, T(u)), suppose that (un) <;;; A(M) and Un --> u as n --> 00. For each n we have J(l(u n )) :::: inf J(M) because J is uniformly continuous on bounded subsets of E. By continuity, (l (un) --> (f/ ( u) and hence J(l(u)) :::: inf J(M). This shows that J remains bounded from below on the positive semiorbit starting at u. Then again [16, Theorem 1.1] and [32, Theorem 3.13] imply that u E g. By the same theorems O+(A) is bounded, and it is relatively compact by the compactness of the flow. 0 3. Existence of Equilibria and Connecting Orbits To formulate existence results on (generalized) connecting orbits, let
K
:=
{u EEl J'(u)
=
o}
denote the set of critical points of J or, equivalently, the set of equilibria of <po We introduce a binary relation on K. Suppose that U1, U2 E K and that there is vEE such that U1 E o:(v) and U2 E w(v). In this case we write U1 >- U2 and say that U1 weakly connects to U2. If 0:( v) = {ud and w(v) = {U2} then we say that U1 connects to U2. We construct weak connections to by means of the
°
14
Lemma 3.1. Suppose that A <;;; Wk and An Dk i= 0 for some k 2:: k o . Then there is u E K n Dk n O+(A) such that u >-- o.
E:
Proof. The manifold Wk is a graph over near 0 of a locally defined map h: E: ---4 Ei: . Let S <;;; Wk denote the graph of a small sphere in E: , centered at O. Now pick Uo E AnDk and consider (un) <;;; A such that Un ---4 Uo as n ---4 00. For n large enough there is tn 2:: 0 with Vn := 'Ptn (un) E S. Since O+(A) is precompact, by Proposition 2.1 (with M = {O}) we may assume that Vn ---4 v in S as n ---4 00. Then a( v) contains u with the desired 0 properties. To prove the existence of an equilibrium in Dk <;;; 8A(O) that weakly connects to 0, for some k 2:: ko, it is therefore sufficient to show that Dk i= 0. Denote
01 a(x) > O}, E 01 a(x) < O}, E 01 a(x) = O},
0+:= {x E
00
0
:=
{x
:=
{x
and F := HJ (0+), considered as a subspace of E by extension with the following is easy to prove:
o. Then
Lemma 3.2. Suppose that Y is a finite dimensional affine subspace of F. Then lim
//uIlE-+<Xl
J(u)
=
-00.
uEY
3.1. Invariant Manifolds and the Comparison Principle First we describe what information one can extract for the geometry of the set A(O) from the comparison principle. Suppose that A < Al (i.e., 0 is linearly asymptotically stable). Then A(O) is an open neighborhood of 0 and, in fact, A(O) = WI. Denote B+ := {u E A(O) 1 'Pt(u) > 0 for some t 2:: O}, B- := {u E A(O) 1 'Pt(u)
< 0 for some t 2:: O},
the basins of upper and lower attraction of 0, and 8A(0)± := 8A(O) n B±.
15
Note that Et\{O} consists of nodal functions. Therefore (1) implies the same for W 2 \{O} and hence that W 2 = A(O)\(B+ U B-). Recall that EI is spanned by the E-normalized positive first eigenfunction el of -6.. Theorem 3.3. [25,32,81] Suppose that A < AI. Then there are an open neighborhood U of 0 in Et and continuous maps h±, h: U -? JR with the following properties:
au i= 0, h - < h < h + on U, h ± = h on au, h is continuously differentiable in U, each of the three sets W 2 , aA(O)+, and aA(O)- does not contain any ordered pair of distinct elements, (e) aA(O)± is the graph of the map v f-+ h±(v)eI' W 2 is the graph of v f-+ h( v )eI' the maps being defined on U, and D2 is the graph of h lau, (J) the basins of upper and lower attraction of 0 satisfy
(a) (b) (c) (d)
+ sel I v E U, = {v + sel I v E U,
JR, h(v) < s < h+(v)},
B+ = {v
S E
B-
s E JR, h-(v) < s < h(v)}.
In other words, A(O) is the set between the graphs of two continuous maps over U <;;; Et, and it is divided into its basin of upper and lower attraction by the graph of a CI-map. All three graphs are unordered. See also the schematic representation in Figure 1.
Figure 1.
Geometry of A(O) if 0 is stable.
16
Since the theorem is partially new in this setting, and to give an idea of the type of arguments used, we provide a
Sketch of the proof of Theorem 3.3. Recall first that by Proposition 2.1 every semiorbit starting in A(O) is relatively compact. Therefore we can repeat most arguments from [25]. In some respects our proof is different though because E is not strongly ordered. First we consider W 2· Suppose there were Ul, U2 E W 2 with Ul > U2' Setting vet) := CPt(Ul) - CPt(U2) we would have vet) ~ 0 for t ~ O. On the other hand, by (2) v(t)/llv(t)IIE would approach a compact set in which only consists of nodal functions. Contradiction! Now suppose that Uo E W 2 and Vo E Tu W 2 were a tangent vector at Uo with Vo > O. Setting vet) := Dcpt(uo)vo we would obtain a contradiction as before, but now using (3). Defining U := piW2 these facts show that W 2 is the Cl-graph of the map v f---+ h( v )el' where h: U ---+ lR is continuously differentiable. Now we claim that h extends continuously to U. Assuming the contrary, suppose that we are given u* E aU, (u~) <:;;; U such that u~ ---+ u* as n ---+ 00, for i = 1,2, and such that with ai := limn-+oo h(u~) E lR U {±oo} we have a2 < al· Pick /3i such that a2 < /32 < /31 < al. We may assume that h(u~) ~ /31 and h(u~) ::; /32 for all n. Setting /3 := /31 - /32 and v~ := u~ + /3ie1 we have that v~ - v; = /3e1 + 0(1) in E as n ---+ 00. Pick t > 0 such that the segment with endpoints u* + /31e1 and u* + /32el is a subset of'Dt (which is open). Set
Ei,
Since cp:
iJ ---+ X
cpt(v~) - cpt(v;)
cpt(u;
is continuously differentiable in the first variable we obtain ---+ 00. Now w » 0 implies that
= W + 0(1) in X as n
+ h(u;)el)
~ cpt(v;") »cpt(v~) ~ cpt(u~
+ h(u;')et}
for large n, contradicting the incomparability of distinct elements in W 2 . This proves the claim. It is clear D2 is the graph of hl&u. Consider two linearly independent elements U1, U2 E F, such that Ul > 0, an let Y denote the 2-dimensional span of Ul and U2. Because no tangent space to W 2 contains a function that is > 0 and because the co dimension of W 2 is 1, Y and W 2 intersect transversally. Moreover, the intersection is a C 1-graph over Y n U, and it is bounded because of Lemma 3.2. This shows that Y n U is bounded and that therefore aU -=/0. We have proved (a), (c), and parts of (d), (e).
17
Since 0 is asymptotically stable, the continuity of 'P implies that a neighborhood of W2 is included in A(O). Consequently, if v E U then v + (h(v) + s)e1 E B+ for small s > o. Moreover, v + (h(v) + s)e1 tJ- A(O) for large s > 0, by Lemma 3.2 and because J(O) = o. These facts show that oA(O)+ f- 0. One proceeds now as in [25], occasionally resorting to arguments as above that use the continuous differentiability in the first argument of 'P: D ----* X. In this way follow (b )-( f). 0 Corollary 3.4. The above argumentation still applies to show the graph and unorderedness property of W 2 if A < A2. Remark 3.5. As already noted, the main difference to the result by PoliCik 25 is that he works under the assumption that the positive cone in state space has nonempty interior. For example, this is true in w~,q (n), with q > N. In this setting he obtains the stronger result that hand h± are globally Lipschitz continuous. Moreover, he also considers the case A = AI, i.e., 0 is a degenerate equilibrium. The condition in [25] that all positive semiorbits are bounded is replaced by Proposition 2.1 in our setting. The proof of the unordered ness of W 2 could also be effectuated by using [81, Remark 2.5] and [25, Lemma 3.2]. A result of Matano 102 ,103 on the existence on connecting orbits between ordered equilibria (when there is no third equilibrium in the order interval given by the former equilibria) implies that every positive or negative equilibrium in oA(O) has a connecting orbit to o. Combining Theorem 3.3 with Lemma 3.1 applied to A ~ B± and A ~ W 2 therefore yields the Corollary 3.6. If A < Al then there are a positive equilibrium u+ and a negative equilibrium u- that both connect to o. If A < A2 then there is a nodal equilibrium that weakly connects to o. Similar results have been obtained by Quittner,104 but without the information on connections to 0 for the nodal equilibrium. For arbitrary equilibria we obtain the following information on their location relative to A(O): Proposition 3.7. [81] Let u be an equilibrium such that u > v for some v E A(O). Then u ~ o. A corresponding statement holds if u < v. By Theorem 3.3 this shows for example that there is no equilibrium in oA(O)+\(PE u D 2) and oA(O)-\( -PE u D2).
18
3.2. The Definite Homogeneous Case This special case is interesting because the geometry in phase space can be described more completely. In this subsection we consider
A = 0,
g(u) = lul p - 1 u,
a == 1
with p E (1, ps). Some of the results below remain true if one more generally considers g such that g(u)/lul is strictly increasing. Theorem 3.8. [9,105] Ifu E PE\.4(O) then cpt(u) blows up infinite time. Proof. If u is a nontrivial equilibrium then the linear operator L := -6.g'(u) satisfies (Lu, u)P(o) < O. Therefore every nontrivial equilibrium is linearly unstable. Arguing by contradiction, suppose that u E P E\.4(O) and that cpt (u) exists for all time. By the a priori bound in Proposition 2.1 0+ (u) is relatively compact. Arguments similar to those used in the proof of Theorem 3.3 yield s E (0,1) such that v := su E 8.4(0)+. As a consequence, the sets w( u) and w( v) are not empty. Since v < u, results in Smith and Thieme 106 imply that either w(v) < w(u) or w(v) = w(u) is a singleton. In the first case, assume that v* E w( v) and u* E w(u). If M := [v*, u*] is the order interval in E, then O+(M) is relatively compact and J achieves its minimum on w(M) in an element w. Since v* E 8.4(0), v* -=I- 0 and w 2: v* > O. Hence w is linearly unstable, contradicting the fact that w is a local minimum for J in X. In the second case, suppose that w(v) = w(u) = {u*}. Then, as for (2), the arguments in [81, Appendix A and B] apply to the orbits of u and v (shifting the equation to the equilibrium u*) and yield a (finite dimensional) eigenspace V of the linearization - 6. - g' (u *) with Dirichlet boundary conditions, orthogonal in L2(Q) to the space spanned by the signed first eigenfunction of this operator, such that
Here, as in (2), 8 1 V denotes the sphere of radius 1 in V. We obtain a contradiction using the facts that cpt (u) > cpt (v) for t 2: 0 and that all functions in V\{O} are nodal. 0 The proof given here essentially is that of [105]. As noted in [9], the last theorem is implicitly included in Lions. 24 ,107 The nonlinearity considered
19
there is convex, but the results partially apply in P E because 9 is convex in [0,00). Denote the set of initial data whose positive semiorbits blow up in finite time by F := E\g. Lazzo and Schmidt105 then proceed to show that P E has an order decomposition in the sense of Hirsch, consisting of A(O) n P E and of F n P E, the closure of the "set of attraction of 00 in P E". One obtains a threshold result: For every u E P E\ {O} there is 0:( u) > 0 such that su E A(O) if s < 0:( u) and su E F if s > 0:( u). The map 0: is continuous by arguments similar to those used in the proof of Theorem 3.3, c.f. [67]. Denote
N:= {u
E
E\{O} [ J'(u)u = O},
the Nehari Manifold of J,
N+
:=
{u E E [ J'(u)u > O},
and
N-:= {u E E [ J'(u)u < O}.
Put Co : = min J (K\ {O}) = min J (N), the minimal nontrivial critical energy. Then Gazzola and Weth 9 prove some interesting relations between A(O), certain energy ranges of parts of u, and the sets N±: Theorem 3.9. [9,108]
(a) (b) (c) (d) (e) (f)
= A(O), ifu E N+ and J(u) < Co then u E A(O), if u E N- and J(u) < Co then u E F, if [[U[[E :::: J2Co then u E A(O), ifu+,u- EN+ andJ(u+),J(u-)
intg
As in Proposition 3.7 the relation of an initial value to equilibria forces a certain behavior of the corresponding orbit: Theorem 3.10. [9] Suppose that u E K\{O} and vEE.
(a) If u+ #- 0 and v > u then v E F, (b) if u- #- 0 and v < u then v E F, (c) ifu > 0 and -u < v < u then v E A(O). The next result more directly compares A(O), F and N±, using points in E with arbitrarily high energies: Theorem 3.11. [9] For any M > 0 there are u, v, wEE such that
J(u),J(v),J(w) 2': M and u E A(O) nN+ nPE, v E FnN+ nPE, and wE FnN-.
20
It is left open in [9] whether A(O)
<;;;;
N+ or N-
<;;;;
F.
3.3. Invariant Manifolds and the Zero Number As is to be expected, in the one dimensional case one can prove better results on the geometry of superstable manifolds and on the existence of equilibria and connecting orbits. Similarly as in the proof of Theorem 3.3, but now using the zero number, we have Proposition 3.12. Let N = 1. For each k ::::: ko the set W k is a Cl-graph over an open neighborhood of 0 in
E: .
Theorem 3.13. [81] If N = 1 and n = (0,1) then there is a doubly infinite sequence (Uk) <;;;; K (k E Z, jkj ::::: k o ) such that Uk connects to 0, uk E Dlkl\Dlkl+l, ((Uk) = jkj- 1, sign8xuk(0) = signk and J(Uk) ---+ 00 as jkj ---+ 00. Because this setting is more general than that in [81] due to the possibility of mina < 0 we give a Sketch of the proof. Pick 0 < Xl < X2 < 1 such that [Xl, X2] <;;;; n+. Denote by {h,h, ... } <;;;; E the eigenfunctions of -.6. on (Xl,X2) with Dirichlet boundary conditions, extended to n by O. For any kEN denote Yk := [h] and Y k- := [h, 12,···, h-d· Then ((u) = k - 1 for all U E Y k and ((u)::; k - 2 for all U E Yk-. Fix k ::::: max{ ko, 2}. For any u E W k and v E Tu W k we have ((u), ((v) ::::: k - 1. Therefore Yk~l transversally intersects Wk, and M := Yk~l n Wk is a Cl-submanifold of Yk~l' a global Cl-graph over Yk. Moreover, M n Wk+1 = 0 because ((u) ::::: k for all u E Wk+l' Since the codimens ion of W k+l as a submanifold of Wk is 1, together with Lemma 3.2 it follows that Dk \Dk+1 contains two elements, and that their w-limit sets are distinct. This and all the other statements are proved as in the proof of [81, Theorem 3.5]. Note that if u is an equilibrium and u >- 0 then u connects to 0 by results of Zelenjak92 and Matano,93 see also [109]. 0 The existence of infinitely many equilibria has in principle been known in the definite case since the work of Ehrmann llO and Nehari,l1l see also Struwe. 112 Apart from yielding more information on the nature of these solutions, our proof also gives an upper bound for the minimum zero number that appears for nontrivial equilibria.
21
It seems to have been Butler 113 who first considered the one dimensional indefinite elliptic problem, but under periodic boundary conditions. For a fixed interval he shows the existence of one solution. With the goal to construct chaotic motions, the existence of infinitely many solutions of the Dirichlet problem is proved by Terracini and Verzini,114 prescribing the exact number of zeros in intervals where a > O. Here the approach of Nehari is extended to the indefinite problem. This result was subsequently improved by Papini and Zanolin 1l5 using phase plane analysis. The striking fact is that in the indefinite case an equilibrium u E E may have rapid oscillations in any of the components of n+ , while in any given component of n- it can be prescribed if there is no or exactly one zero of u. It would be interesting to prove these results using the parabolic equation (P A ) in the spirit of Theorem 3.13.
3.4. Invariant Manifolds and Linking Theorems The technique we are about to present in this subsection is applied in the case A ::::: A2, i.e., where the nonpositive generalized eigenspace of -.6. - A in E has dimension at least 2, because here the results from Sect. 3.1 do not hold. Then ko > 2 and we wish to prove that Dko -=I- 0 to obtain an equilibrium in Dko with weak connection to 0 by Lemma 3.1. It follows as in Theorem 3.3( d) that all functions in Wko \ {O} change sign; therefore the equilibria constructed here are nodal. In the definite case we have
Theorem 3.14. [81] Suppose that mina > 0, and that (mina)G(u) :::::
Ak
-1 0
2
A 2 u
for all u E R
(6)
Then there is a nodal equilibrium u* that weakly connects to O.
Proof. The condition (6) implies that J ::; 0 on Eko' Pick w E Eto with !lwll = 1 and set Y := Eko EEl [wJ. By Lemma 3.2 there is R > 0 such that J(u) ::; 0 if u E Y and Ilyll : : : R. Define
M
:= { v
+ sw I v
E
E k o-, !Iv + sw!l ::; R, s ::::: 0 }
and let Mo denote the boundary of M in Y. Hence J ::; 0 on Mo. Let S denote the lift to Wko of a small sphere in Eta centered at 0, using that Wko is a graph over Eto near O. Then M and S link in the following sense: Any continuous 't/J: M ----> E with 't/JIMo = id Mo intersects S. Moreover inf J(S) > O.
22
Consider the continuous semiflow rp one obtains when stopping 'P at nonpositive energy levels of J. For a sequence tn -+ 00 there are Un E M with rptn (un) E S. Passing to a subsequence we may assume, since M is compact, that (un) converges to some u E M. It follows that u E Dk o ' and hence Lemma 3.1 applies. 0 For the indefinite case it is difficult to give explicit conditions on a and 9 such that the above linking construction yields a result. Nevertheless, one can instead simply assume integral conditions to hold on the respective generalized eigenspaces, as has been done by Grossi, Magrone and Matzeu. 116 Combining this idea with the technique used above we obtain Theorem 3.15. Suppose that A E [Ak-l, Ak), that
In
a(x)G(u) dx 2: 0
for all u E
E;
and that there is W E Et\{O} such that J(u) -+ 00 as [[ull -+ 00 and u E E; EEl [w]. Then there is a nodal equilibrium u* that weakly connects to
o. Generally we can always prove an existence alternative, using a local linking type approach: 117 Theorem 3.16. [32] If A > A2 and A rJ- a( -.6.) then there is a nontrivial equilibrium u* such that either J(u*) < 0 and 0 >- u* or J(u*) > 0 and u* >- O.
Similar results are proved in [33,118] in the variational setting, also using local linking arguments, but without the information on connecting orbits for 'P. As usual, an odd nonlinearity gives an infinity of equilibria: Theorem 3.17. [32,81] Suppose that 9 is odd. Then there is a sequence (un) of nodal equilibria of (P A ) that weakly connect to 0, and that satisfies J(u n ) -+ 00 as n -+ 00. Sketch of proof. For every Wk with k 2: ko, using a linking argument as in the proof of Theorem 3.14 one shows that Dk -I- 0. Here the theorem of Borsuk-Ulam yields the linking structure. Therefore every Dk contains an equilibrium that weakly connects to 0, by Lemma 3.1. Finally, Eq. (4) together with the Palais-Smale property of J implies that J is not bounded on the set of these equilibria. 0
23 In the definite case with odd g Quittner 104 also proves the existence of infinitely many equilibria in aA(O). In the indefinite case the existence of infinitely many equilibria is proved by Alama and Tarantello 45 and Magrone and MatalonL 119 The latter reference also treats perturbations from symmetry. None of these articles proves that the obtained equilibria are nodal.
3.5. Order Intervals, Sub- and Supersolutions In this subsection we will consider order intervals formed by signed stationary sub- and supersolutions, and their interaction with the unstable manifold of O. Let us denote A+
:=
{A
E
IR I (E A) has a positive solution},
A - := {A E IR I (E A) has a negative solution}, and
A± := supA±. By replacing g(u) with -g(-u) if necessary we assume throughout that
A- -:; A+; this facilitates the presentation. The basic properties of A± are given in the following Lemma, whose proof can be found in [32]. Lemma 3.18. Al -:; A+ < 00 and (-oo,A+) <;;; A+. Moreover, A+ E A+ if A+ > AI. Given g with (G) one can find a sign changing function a satisfying conditions (A) and (/) such that A+ is arbitrarily large. Similar statements hold for A- . Some of these facts have been known for a long time. The boundedness of A+ was proved by Ouyang,120 while A+ 2: Al simply follows from Corollary 3.6. That A+ can be arbitrarily large seems to be new in [32]. In a restricted setting more is known: Theorem 3.19. [45] Suppose that g(u) = uP for u > O. Then A+ > Al holds if and only if
in
a(x)ef+l dx < O.
(7)
24
Note that in [45] more general functions 9 are admitted. Similar or related results were obtained in [41,46,51,121,122]. This fact explains why, if A > AI, the existence of positive equilibria is generally only expected in the indefinite case.
Theorem 3.20. [123] Assume g(u) = lulp-Iu and (7). Suppose that positive solutions of (E.\) possess uniform Leo a priori bounds for A in compact subsets of R Denote by AI(Q\Q+) the first Dirichlet eigenvalue of-~ on Q\Q+. Then Al < A+ < AI(Q\Q+). Moreover, for each A E (AI,-A+j Eq. (E.\) possesses a unique linearly stable positive solution. If A E (AI, A+) then this solution is linearly asymptotically stable. Observe that if A > A+ then every orbit of
Theorem 3.21. [126] Suppose that g(u) = uP for u
IW
A := { W E X
B := {w E C;(Q)
> O. Denote
~ 0}
IW
2: O}
and L(u v) := ,
In \1u· \1(~) dx f
In
(VP+l) d up-1 X
for u E A and v E B. Then A+ = sup inf {L(U, v) uEAvEB
IIrn a(x)vP+l dx > o} =
inf su p { L(u, v) vEBuEA
IInra(x)v p+ dx > o}. l
This result is proved without assuming a priori estimates (not even for positive orbits). It therefore extends to all p E (1,ps). From Section 3.1 we already know that if A < Al then there are three nontrivial equilibria (one positive, one negative, and one nodal) that (weakly) connect to O. We now present multiplicity results for the case A> AI:
25 Theorem 3.22. [32]
(a) If Al < A < A+ then there exist at least three nontrivial equilibria: two positive and one nodal. (b) If Al < A < A-then there exist at least five nontrivial equilibria: two positive, two negative, and one nodal. (c) If A2 < A < A+ then there exist at least four nontrivial equilibria: two positive and two nodal. (d) If A2 < A < A- and A f/. O"(L) then there exist at least seven nontrivial equilibria: two positive, two negative, and three nodal. Sketch of Proof of (d). We assume for simplicity that there are only finitely many equilibria. First we prove the existence of two positive equilibria. Pick a positive solution u+ of (Ex) for some XE (A, A-). Then u+ is a strict positive supersolution of (EA)' that is, -~u+ 2:: AU+ +a(x)g(u+). For small c: > 0 the function ~+ := c:e1 is a strict positive subsolution of (E A ) such that ~+ < u+. Then there is a positive equilibrium which is the minimum of J on w([~+, u+]). The set [~+, u+] strongly attracts a neighborhood and therefore A([~+, u+]) is open. Moreover, O+(
ut
ut
uf
Remark 3.23. In Theorem 3.22 one can also give some information on the existence of connecting orbits between equilibria. Also, it suffices to assume in (d) that 0 has some nontrivial critical group in dimension greater than one. Dancer 127 has more information on the critical groups in this setting. Multiplicity results for equilibria in the indefinite case have been given in [45,48,121]. None of them contain results on the existence of nodal equilibria when also positive equilibria exist. On the other hand, if A ¢: 0"( -~) then in [33] the existence of a nontrivial equilibrium is shown. This proves the
existence of a nodal equilibrium if in addition A > A+, since then no signed
26 equilibria exist. The more general elliptic problem -~u = AU + a+(x)lulq-1u - a_(x)luIP-1u { U(X) = 0
xEn xE
an
(8)
with 1 < q < p < Ps and a_, a+ 2': 0 was considered by Alama and Tarantello 47 and Chang and Jiang. 34 In both articles multiplicity results are obtained. In the latter, the existence of a nodal solution is proved under the assumption of an a priori bound for positive solutions. Chang and Jiang 52 also apply the parabolic flow method presented here to problem (8). As at the end of Section 3.3 we consider the existence of solutions with prescribed behavior in n±. In G6mez-Reiiasco and L6pez-G6mez 123 it is conjectured (backed up by numerical computations) that, at least for large - A and in one space dimension, one should expect the existence of a multitude of positive solutions that have most of their mass in n+, distinguished by the number and locations of masses in the components of n+. In [17,18] this conjecture is interpreted in the setting of systems that model the population dynamics of two species in a heterogeneous habitat. The conjecture is proved by Gaudenzi, Habets and Zanolin 128 ,129 in a slightly different setting, not using A as a parameter but instead considering g(u) = (a+(x) -IJa_(x))luIP-1u, where a± > 0 on n± and a± = 0 on n'f. If n is the number of components of n+ then for large J.l > 0 they obtain 2n - 1 positive solutions, given by the combinatorial possibilities of distributing a positive number of masses on the components of n+. It would be interesting to understand the parabolic dynamics in this setting. Moreover, can one construct nodal equilibria in the spirit of the one dimensional case (see the end of Section 3.3), and what are the characteristics of the corresponding parabolic dynamics?
Acknowledgment I would like to thank Professor Yihong Du for inviting me to contribute to this volume.
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31
A NOTE ON REACTION-DIFFUSION SYSTEMS WITH SKEW-GRADIENT STRUCTURE Chao-Nien Chen
Department of Mathematics, National Changhua University of Education, Changhua Taiwan, ROC E-mail: [email protected] Tzyy-Leng Horng
Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan, ROC E-mail: [email protected] Daniel Lee
Department of Mathematics, Tunghai University, Taichung, Taiwan, ROC E-mail: [email protected] Chen-Hsing Tsai
General Education Center, Chung Chou Institute of Technology, Changhua Taiwan, ROC E-mail: [email protected] Reaction-diffusion systems with skew-gradient structure can be viewed as a sort of activator-inhibitor systems. We use variational methods to study the existence of steady state solutions. Furthermore, there is a close relation between the stability of a steady state and its relative Morse index. Some numerical results will also be disussed.
1. Introduction
In this note we consider reaction-diffusion systems of the form MIUt M2Vt
= DI6.u + FI(u, v), = D 2 6.v - F 2 (u, v),
(1.1)
x E 0, t > O.
(1.2)
Here 0 is a smooth bounded domain in IRn, u(x, t) is an ml-dimensional vector function, v(x, t) is an m2-dimensional vector function, M I , M 2 , DI
32
and D2 are positive definite matrices, and there exists a function F such that \1 F = (FI' F2)' Such systems can be viewed as a sort of activatorinhibitor systems. A well-known example is
Ut
TVt
= dl~U + f(u) - v, = d2~V
+ au -
,v,
(1.3) (1.4)
where d l , d 2 , a, " T E (0,00) and f is a cubic polynomial. The case of d2 = has been considered as a model for the Hodgkin-Huxley system 13, 22 to describe the behavior of electrical impulses in the axon of the squid. More recently, several variations of this system appeared in neural net models for short-term memory and in nerve cells of heart muscle. As in [29], (1.1)-(1.2) will be referred as a skew-gradient system in which a steady state is a critical point of
°
1>(u, v)
I
= [ -(Dl \1u, \1u) n 2
1
- -(D2 \1v, \1v) - F(u, v)dx. 2
(1.5)
A steady state (u, v) is called a mini-maximizer of 1> if u is a local minimizer of 1>(-, v) and v is a local maximizer of 1>(u, -). It has been shown 29 that non-degenerate mini-maximizers of 1> are linearly stable. This result gives a natural generalization of a stability criterion for the gradient system in which all the non-degenerate local minimizers are stable steady states. A remarkable property proved in [29] is that any mini-maximizer must be spatially homogeneous if n is a convex set. This kind of results have been established by Casten and Holland5 and Matano 20 for the scalar reactiondiffusion equation, and generalized by Jimbo and Morita 15 and Lopes 19 for the gradient system. In case n is symmetric with respect to Xj, Lopes 19 showed that a global minimizer of gradient system is symmetric with respect to Xj; while Chen7 obtained parallel results for the global mini-maximizers in the skew-gradient system. In connection with calculus of variations, there is a close relation between the stability of a steady state of skew-gradient system and its relative Morse index. Based on this idea, some stability criteria for the steady states of (1.1)-(1.2) are illustrated in section 2. In section 3, variational arguments are used to study the existence of steady states and their relative Morse indices. Section 4 contains numerical investigation of skew-gradient systems.
33
A particular example to be studied is Ut
T2Vt T3Wt
= d1u xx + f(u) - v = d 2 v xx + U - /2V, = d 3 wxx + u - /3W,
w,
which served as a mode1 4 for gas-discharge systems.
2. Stability Criteria Let E be a Hilbert space. For a closed subspace U of E, Pu denotes the orthogonal projection from E to U and U.l denotes the orthogonal complement of U. For two closed subspaces U and W of E, denoted by U '" W if Pu - Pw is a compact operator. In this case, both W nUl. and W.l n U are of finite dimensional. The relative dimension of W with respect to U is defined by
dim(W, U)
= dim(W nul.) - dim(W.l n U).
(2.1)
If A is a self-adjoint Fredholm operator on E, there is a unique A-invariant orthogonal splitting
with E+(A), E_(A) and Eo(A) being respectively the subspaces on which A is positive definite, negative definite and null. For a pair of self-adjoint Fredholm operators A and A, it will be denoted by A '" A if E_ (A) '" E_ (A). In this case, a relative Morse index i(A, A) is defined by
(2.2) We refer to [1] for more details of relative Morse index. For a critical point (iL, v) of
-D
D= ( o 1
and h be the k x k identity matrix. For a gradient system, it is known that a non-degenerate critical point with non-zero Morse index is an unstable
34
steady state. The next theorem gives a parallel result for the skew-gradient system.
Theorem 1. Suppose i( -Q, ~II(U, v)) i= 0 and dimEo(~II(u, v)) = 0, then for any positive definite matrices Ml and M 2, (u, v) is an unstable steady state of (1.1)-(1.2). In [29] Yanagida pointed out an interesting property that a nondegenerate mini-maximizer of ~ is always stable for any positive matrices Ml and M2 given in (1.1)-(1.2). An interesting question is whether there exist steady states with stability depending on the reaction rates of the system. Let P+ and P- be the orthogonal projections from E to E+(Q) and E_(Q) respectively. Define Wo = M-! (Dtl- \1 2 F(u, v))M-!, 'l/Jl = P-WoP- and 'l/J2 = P+WoP+. Set m = ml + m2, f) = H2(f2,lR m ),
. ('l/Jlz, z) £2 Pi('l/JI) = zE:D mf IIP- z 112£2
(2.3)
('l/J2 Z ,Z)£2 Ps('l/J2) = zE:D sup IIP+ 112 . Z £2
(2.4)
and
Theorem 2. Assume that i(-Q,~II(U,V)) = 0 and Then (u, v) is stable if Pi('l/JI) > Ps('l/J2).
dimEo(~II(u,v)) =
o.
= vlan
=
Remark. In case we treat the Dirichlet boundary condition ulan 0, f) is replaced by H 2(f2,lR m )nHJ(f2,lRm ). The proofs of Theorem 1 and Theorem 2 can be found in [8].
3. Applications of Theorem 1 and Theorem 2 In dealing with a strongly indefinite functional ~, a critical point theorem established by Benci and Rabinowitz 3 can be used to obtain steady states of (1.1)-(1.2).
Theorem 3. Let E be a separable Hilbert space with an orthogonal splitting E = W+ EB W_, and Br = W~ E E, II~II < r}. Assume that ~(~) = ~(A~,~) + b(~), where A is a self-adjoint invertible operator on E, b E C 2 (E,IR) and b' is compact. Set S = oBp n W+ and N = {~- + sel~- E Br n W_ and s E [0, .Ill}, where e E OBI n W+, r > 0 and R> P > o. If ~ satisfies (PS)* condition and sUPaN ~ < infs~,
35 then iP possesses a critical point [ such that infs iP ;::: iP([) ;::: Moreover, if W_ rv E_, then
i(A, iPJ/([)) ::; dim(W_, E_)
SUPaN
iP.
+ 1::; i(A, iP"([))
+ dimEo(iPJ/([)).
(3.1)
Remark. (a) See e.g. [2, 8J for the definition of (PS)* condition. (b) The index estimates (3.1) were obtained by Abbondandolo and Molina. 2 In a demonstration of using Theorem 3 to study the existence and stability of steady state solutions, we consider a perturbed FitzHugh-Nagumo system in the first example:
= d1fl.u + J(u) - v,
(3.2)
= d2fl.v + u - "'(v - h(v).
(3.3)
Ut
TVt
A steady state of (3.2)-(3.3) is a critical point of
iP(u, v)
=
1
dl 2 d2 2 -[Vu[ - -[Vv[ - F(u,v)dx, 2
n 2
where
F(u, v)
=-
(~u4 -
(3; 1 u3
+ ~u2)
-
UV
+ ~v2 + H(v),
(3.4)
v
(3 E (0, ~) and H(v) = Jo h(y)dy. It is assumed that "'( > 9(2(32 - 5(3 + 2)-\ and h satisfies the following condition: (hI) hEel, h(O) = h'(O) = 0 and yh(y) ;::: 0 for y E R Define
Let
1
ILt = 2' [(d 1 -
d2»'k - (1'(0)
+ "'() + J((d 1 + d2»'k -
d2».k - (1'(0)
+ "'() -
J'(O)
+ "'()2 + 4]
J((d 1 + d2»..k - J'(O)
+ "'()2 + 4],
and
ILk = ~ [(d 1 -
where {-Ak} are the eigenvalues of the Laplace operator and {¢d are the corresponding eigenfunctions. By straightforward calculation Aet ¢k = ILt et ¢k and Ae k¢k = ILk ek¢k, where
et
=
(I, ~[J((dl + d2)Ak - J'(O) +
"'()2
+
4- [(d
1
+ d2)Ak -
J'(O)
+ "'(ll)
,
36
ek = (1,
~1 [(d 1 + d2»'k -
1'(0) + 1 + v'((d 1 + d2»'k - 1'(0) + 1)2 +
4]) .
It is clear that ILt > 0 and ILk < 0 for all kEN. Let E+ = EBk:,1 V/ and E_ = EBk=lVk-, where V/ = {s
for
ZI, Z2
E E. As an application of Theorem 3, we have the following result.
Theorem 4. Let B R be a ball in lRn with radius R. If n contains a ball BR with R being sufficiently large, then there exists a steady state (u, v) of (3.2)-(3.3), and i(-Q,q,II(U,V)):::; 1:::; i(-Q, q,1I(U, v))
+ dimEo(q,II(u, v)).
In view of Theorem 1, (u, v) is unstable if it is a non-degenerate critical point of q,. More details can be found in [8]. We now turn to some examples to seek stable steady states of skewgradient systems. Consider Ut
TVt
= D.u -
= D.v + 2v + U
v,
(3.5)
IV Iv.
(3.6)
U -
-
Straightforward calculation gives
A=(-~+l D.~2)' ILt = ~(3 + v'(2 Ak - 1)2 + 4) and ILk = ~(3 - v'(2Ak - 1)2 + 4). It is clear that ILt > 0 for all kEN. Suppose n is a bounded domain in which the eigenvalue distribution of the Laplace operator (under homogeneous Dirichlet boundary conditions) satisfies the following property: 1 Al < 2(/5+ 1) < A2 :::; A3 :::; ... :$ Ak ...
Then it is easily seen that i( -Q, A) = -1.
ILl >
0, and ILk < 0 if k 2: 2. It follows that
Theorem 5. There is a non-constant steady state (u, v) of (3.5)-(3.6). Moreover, if dim(q,II(u, v)) = 0 and T 2: ~+~;, then (u, v) is stable.
37 In the next example, consider (1.3)-(1.4) with f(u) (j = 1. Suppose there is a j E N such that if d1)...j
+d
)...1 2j+,
< a < inf {d1)...k + d )... 1
2k+,
°
=
au - u 3 and
Ik E N\{j}}.
(3.7)
°
By direct calculation J-lj < and J-lt > for k E N\ {j}. Also, J-lk < 0 for all kEN. Hence i( -Q, A) = 1. Applying Theorem 3 yields a steady state (il, v) of (1.3)-(1.4). Furthermore,
i( -Q, -~II(il, v)) ::;
°: ; i( -Q,
-~II(il, v))
+ dimEo(~1I (il, v)).
°
This implies that i( -Q, -~II(il, v)) = if (il, v) is a non-degenerate critical point of ~. Then by Theorem 2, (il, v) is stable if T < ;. In case of dealing with homogeneous Neumann boundary conditions, (il, v) is a spatially inhomogeneous steady state if (3.7) holds for j 2: 2. In other words, there exists a stable pattern for (1.3)-(1.4). For the FitzHugh-Nagumo system, the steady state solutions satisfy d1b.u + f(u) - v = 0, d2 , -b.v + u - -v (j (j = 0,
(3.8) (3.9)
!).
where f(u) = (1- u)(u - j3)u, j3 E (0, If.c = (j-1(-d 2 b. + ,)-1 under homogeneous Dirichlet (respectively Neumann) boundary conditions, then for any critical point il of
(il, .cil) is a steady state of FitzHugh-Nagumo system. In view of the fact
that (j
in
u.cudx =
in
d2 1V'v 12 + ,v2 dx, it is easily seen that 'ljJ is bounded
from below. In addition to minimizers, the Mountain Pass Lemma has been used to obtain non-trivial solutions9-11, 17, 21, 24, 28, 32 of (3.8)-(3.9). Let u be a critical point of'ljJ. Straightforward calculation yields 'ljJ"(U)
=
-b. +.c
-
f'(u),
where 'ljJ" is the second Frechet derivative of 'ljJ and the Morse index of u will be denoted by i*('ljJ"(U)). On the other hand, (u, .cu) is also a critical point of
r
~(u, v) = in [d211 V'u I2 -
d
21 V'u1 2
2(j
+ uv -
2'(j v ior f(f,)df,] dx. 2
-
38
Proposition 1. If u is a critical point of'ljJ and v
dimEo( 'ljJ" (u))
= .cu, then
= dimEo( 4>" (u, v))
and
i*('ljJ"(u))
= i(-Q, 4>"(u, v)).
We refer to [8] for a proof of Proposition 1. For a critical point u obtained by the Mountain Pass Lemma, it is known 6 that
i*('ljJ"(u)) ~ 1 ~ i*('ljJ"(u))
+ dimEo('ljJ"(u)).
Then by Proposition 1
i(-Q, 4>"(u,.cu)) ~ 1 ~ i( -Q, 4>"(u,.cu))
+ dimEo(4)''(u, .cu)).
Thus if dimEo ('ljJ" (u)) = 0, it follows from Theorem 1 that (u,.cu) is an unstable steady state of (1.3)-(1.4). Let ~l = P-(DI:::. - \1 2F(u, v))P-, ~2 = P+(DI:::. - \1 2F(u, v))P+, (3.10)
and (3.11)
°
Theorem 6. Assume that i(-Q,4>"(u,v)) = and dimEo(4)''(u,v)) Then (u, v) is stable if one of the following conditions holds: (i) Pi(~d > 0, Ps(~2) 2: and
= 0.
°
PS(~2)
< IIM;111-11IM11I- 1.
Pi('ljJd
(ii) Pi(~d ~ 0, Ps(~2) <
°and
Pi(~l)
< IIMl11l-11IM211-1.
Ps( 'ljJ2) Theorem 6 directly follows from Theorem 2. We refer to [8] for the detail. If u is a non-degenerate minimizer of'ljJ and v = .cu, then Proposition 1 implies that i( -Q, 4>(u, v)) = 0. Notice that
DI:::. - \1 2 F(u, v)
=
-dll:::. - f'(u) (
1
1 d2 1:::. _
1 ).
IJ
IJ
39 Since f'(~) = -3~2 + 2((3 + 1)~ - (3 S ((32 - (3 + 1)/3, it easy to check that PiC(fd = Pi(-d1 D.- f'(u)) ~ d1 >'l- ((32-f+l) and P8(~2) = P8(~D.-;) S -(d2 Al + ,)/<7, where Al < A2 S A3 S ... S Ak < ... are the eigenvalues of -D.. If Pi(~d SO and T < C1«{323~~2:;)-+:..-r;dl)..d' condition (ii) of Theorem 6 holds and consequently (u, v) is a stable steady state of (1.3)-(1.4). 4. Numerical Results
We report some numerical work on the skew-gradient systems, and compare with the theoretical results. 4.1.
We start with the following reaction-diffusion system:
(4.1) = d1 u xx + u(u - (3)(1 - u) - v - W, (4.2) T2Vt = d2v xx + U -,2V, (4.3) T3Wt = d3 wxx + U -,3W, X E (0,1), t > 0. where (3 = 0.3, = 1, = 20, and the homogeneous Neumann boundary Ut
,2
,3
conditions will be under consideration. In (4.1)-(4.3), u can be viewed as an activator while v and W act as inhibitors. In view of the theoretical results mentioned in the previous sections, we look for the pattern formation for (4.1)-(4.3) in case the diffusion rate of the activator is small (d 1 = 10- 6 ). By taking d2 = 1 and d3 = 10- 6 , various types of spatially inhomogeneous steady states have been observed through numerical calculation.
0.9 0.8
I
,..,
ul
0.7 0.6 0.5 :J
0.4
0.3 0.2
o. 1
-0.1
o
0.2
Fig. 1.
0.4
x
0.8
0.8
Solution profile of u.
40 0.045 0.0'
I-
---w ~
I I
I I I
0.035
0.03 0.025
~
0.02 0.Q15 0.01
0.005 ____________ 1
"" _ _ _ _ _ _ _ _ _ _ _ _
-0.0050:-----,0:':.2:-----:::0.':-.----,0:':.•:-----::0.':".-----:
x
Fig. 2.
Profiles of v and w.
0.9r====:---~---~--~_--__,
0.8
I
ul
0.7 0 .• 0.5 ~
0.4
0.3
0.2 0.1
o
-
---J
-O.10'----0~.2:----:0~ .•---0:':.•:----:0':.•
x
Fig. 3. 0....
0.0. 0.035 0.03 0.025
~
0.02 0.015 0.01
0.005
Solution profile of u.
j ... --w v !I: ~, I I I
I I I
I
I
I
I
I
I
I I
I I
,, ,, -!-.' ,
I
, I I
I I I
,, I
I _ _ _ _ II
,t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-O·0050~----;'0.:-'-0:':.2:---:0~.3:--:0':-.4--:05:--0:':.•:---:0:':.7:--:0.':-.--:0.•:-~
x
Fig. 4.
Profiles of v and w.
41 o. 9 o.
•
I
ul
o.7 o.6 o. 5 ::>
o. 4
o.3 o. 2 o.
,
0_ -0.
,
'---
0.2
Fig. 5.
0.4
0.6
0.8
Solution profile of u.
O.045r:====---~---~----:,~,----,
0.04
I___ ~II~
0.035
f.
I.
II
I,
II
I, I)
II I,
I,
..
'I
0.03
0.025
.1
'
"
'
" "
0.02
" " O.015rr_ _~,~,; ________
--:-i---j
0.01
I I
" " "
0.005
" _______________ '
____ I
.1
~.OO50~--~0~.2~--70.~4----:0~.6~--70.7.--~
x
Fig. 6. o.
•I
o. 8
Profiles of v and w
ul
o. 7 O.6 O. 5 ::>
O. 4 O. 3 O. 2
O.
,
0
-0.
,
'0.2
Fig. 7.
0.4
x
0.6
0.8
Solution profile of u.
42 0.045
0.04
...
---w ~
0.035 0.03
0.025
~
" " "
.."
0.02
,I
" "
" """ "
II
,I II II ,I ,I
I'
I'
0.015
0.01
II I I
II .1
It
II
II I
II II I I
.t
0.005
o ___________ ,I
1_, 1_ _ _ _ _ _ _ _ _ _ _
_0.OO5L---~--~--~--~--..J
o
0.2
Fig. 8.
0.4
x
0.6
0.8
Profiles of v and w.
In Figures 1 and 3, there is one peak on the profile of u; the one in Figure 1 is symmetric with respect to the spatial variable, while the other is not. We found also instances of steady states with two peaks on the profile of u; but the distance between peaks can be different. We remark based on numerical observation that, with 72 = 73 = 10- 4 , such inhomogeneous steady states are stable under the flow generated by (4.1)-(4.3). Moreover, the solution profiles tell that w is roughly equal to 1';lU in magnitude. We next turn to the case when both inhibitors v and ware acting with large diffusion (d 2 = d3 = 1). As show in Figures 9-10, the pulse (or peak of u) becomes wider. The fact that 1'3 > 1'2 results in v > w.
2
I
ul
1
r--1
o. 8 o. 6 o. 4
o2
-0.2
o
0.2
Fig. 9.
0.4
x
0.6
0.8
Solution profile of u.
43
r:::::=vl
0.045
0.04~ 0.035
0.03 0.025
~
0.02 0.015 0.01
0.005
-0.OO5L..--~-~--~_ _~_ _---1
o
0.2
0.4
Fig. 10.
x
0.6
0.8
Profiles of v and w.
Keeping d3 = 1 and reducing d2 to 10- 1 , we obtain a stable steady state with rather different profiles as shown in Figures 11-12.
4.2.
In this subsection we come back to the reaction-diffusion system Ut TVt
=
U xx -
v,
U -
= Vxx + 2v + U
Ivlv, x E (0,3), t > 0, -
u(O, t) = v(O, t) = u(3, t) = v(3, t) = O.
1.2
I
ul
0.8
0.6 :>
0.4 0.2
-0.2 0
0.2
-
Fig. II.
~
L--0.6
0.'
0.8
X
Solution profile of u.
44 0.035 0.03
E;] ---w
0.02
~
/'-.
/
0.025
0.015 0.01
o
---------*~----',~--------
_0.OO5L---~-~--~--~----l
a
u
Fig. 12.
u
x
u
u
Profiles of v and w.
Times=20 '1":0.1
0.3 0.2 ........... . 0.1
-0.1
-0.2 00 2
a a
Fig. 13.
Flow of u with
T
= 0.1.
As we know from Theorem 5, the choice of T = 0.1 leads the flow converging to a non-constant steady state (Figure 13). The behavior in the phase plane of the state variables, at the midpoint of the domain (x = 1.5), exhibits a spiral-inward convergence (Figure 14). On the other hand, we conjecture that such a non-constant steady state become unstable if the value of T is taking much smaller. Indeed, when T = 0.005, we observed a time-periodic attractor (Figures 15-16). The convergence history of the two calculated state variables is recorded in Figures 17-18, which strongly suggests the existence of a stable timeperiodic solution. The change of stability seems to result from a Hopf bifurcation and deserves further investigation.
45 Times=20 t=O.l 0.1
0.05
01
u
Fig. 14.
The trajectory of (u (1 ., 5 t) , v(l.5, t)). Tlmes=6 ~.005
.. ".
.. -...........
........•
... .... '.~ ..... . ........ "
-.
0.4 0.2
-0.2
o
Fig. 15.
0
Flow ofu
WI·th T
0.05
0.1
= 0.005.
0.15
0.2
0.25
u
Fig. 16.
The trajectory of (u (1 ., 5 t) , v(l.5, t)).
46 O."'r-_~_ _--,.::.The::.:~::::""""'=;.:::Of::.:"':..:_="~_~_ _--, 0.04 0.035 0.03
Fig. 17.
Historic space-accumulated It-difference of u.
" Fig. 18.
Historic space-accumulated It-difference of v.
Acknowledgments Research is supported in part by the National Science Council, Taiwan, ROC.
References 1. A. Abbondandolo, "Morse Theory for Hamiltonian Systems", Chapman Hall/CRC Research Notes in Mathematics, 425 (2001). 2. A. Abbondandolo and J. Molina, Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems, Calc. Var. 11 (2000), 395-430.
47 3. V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273. 4. M. Bode, A. W. Liehr, C. P. Schenk and H. -G. Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a threecomponent reaction-diffusion system, Physica D 161 (2002), 45-66. 5. R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), 266-273. 6. K. C. Chang, "Infinite Dimensioanl Morse Theory and Multiple Solution Problems", Birkhauser. Basel, 1993. 7. C. -C. Chen, Symmetry properties for the mini-maximizers of skew-gradient systems, Private communication. 8. C. -N. Chen and X. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations 33 (2008), 189-208. 9. E. N. Dancer and S. Yan, Multipeak solutions for the Neumann problem of an elliptic system of FitzHugh-Nagumo type, Proe. London Math. Soc. 90 (2005), 209-244. 10. E. N. Dancer and S. Yan, A minimization problem associated with elliptic systems of FitzHugh-Nagumo type, Ann. IHP-Analyse Nonlineaire 21 (2004), 237-253. 11. E. N. Dancer and S. Yan, Peak solutions for an elliptic system of FitzHughNagumo type, Ann. Se. Norm. Super. Pisa Cl. Sci. 2 (2003), 679-709. 12. D. G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836-849. 13. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961), 445-466. 14. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39. 15. S. Jimbo and Y. Mortia, Stability of nonconstant steady-state solutions to a Ginzburg-Landau equation in higher space dimensions, Nonlinear Anal. 22 (1984), 753-770. 16. H. Kielhofer, Stability and semilinear evolution equations in Hilbert space, Arch. Rational Meeh. Anal. 57 (1974), 150-165. 17. G. Klaasen and E. Mitidieri, Standing wave solutions for system derived from the FitzHugh-Nagumo equations for nerve conduction, SIAM. J. Math. Anal. 17 (1986), 74-83. 18. G. Klaasen and W. Troy, Stationary wave solutions of a system of reactiondiffusion equations derived from the FitzHugh-Nagumo equations, SIAM J. Appl. Math. 44 (1984), 96-110. f9. O. Lopes, Radial and nonradial minimizers for some radially symmetric functionals, Electron. J. Differential Equations 1996 (1996), 1-14. 20. H. Matano, Asymptotic behaviour and stability of solutions of semilinear elliptic equations, Pub. Res. Inst. Math. Sci. 15 (1979), 401-454. 21. H. Matsuzawa, Asymptotic profiles of variational solutions for a FitzHugh-
48
22. 23. 24.
25.
26. 27.
28. 29. 30.
31.
32.
Nagumo type elliptic system, Differential Integral Equations 16 (2003) 897-926. J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Pmc. I. R. E. 50 (1962), 2061-2070. Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J. Appl. Math. 49 (1989), 481-514. Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations 188 (2003), 110-134. P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations", C.B.M.S. Reg. Conf. Series in Math. No. 65, Amer. Math. Soc., Providence, RI, 1986. C. Reinecke and G. Sweers, A positive solution on ]Rn to a equations of FitzHugh-Nagumo type, J. Differential Equations 153 (1999), 292-312. C. Reinecke and G. Sweers, Existence and uniqueness of solutoin on bounded domains to a FitzHugh-Nagumo type elliptic system, Pacific J. Math. 197 (2001), 183-211. X. Ren and J. Wei, Nucleation in the FitzHugh-Nagumo system: Interfacespike solutions, J. Differential Equations 209 (2005), 266-301. E. Yanagida, Mini-maximizers for reaction-diffusion systems with skewgradient structure, J. Differential Equations 179 (2002), 311-335. E. Yanagida, Standing pulse solutions in reaction-diffusion systems with skew-gradient structure, J. Dynam. Differential Equations 14 (2002), 189205. E. Yanagida, Reactoin-diffusion systems with skew-gradient structure, IMS Workshop on Reaction-Diffusion Systems, Methods Appl. Anal. 8 (2001), 209-226. J. Wei and M. Winter, Clustered spots in the FitzHugh-Nagumo system, J. Differential Equations 213 (2005), 121-145.
49
CHANGE OF ENVIRONMENT IN MODEL ECOSYSTEMS: EFFECT OF A PROTECTION ZONE IN DIFFUSIVE POPULATION MODELS Yihong Du* Department of Mathematics School of Science and Technology University of New England Armidale, NSW 2351, Australia E-mail: [email protected]
Abstract: In this paper, we report some recent theoretical work of Du-Shi14, Du-Liang 12 and Du-Peng-Wang 13 that examines the effect of environmental changes in three well-known diffusive population models, each modeling the interaction of two population species in a common habitat. For each model, we consider the situation that one of the two species is endangered and a simple protection zone is created in the habitat for the endangered species. The focus is on the effect of the protection zone on the dynamics of the two species systems. We demonstrate that, though the effects of the protection zone on the dynamical behavior of the three systems are significantly different from each other, they all share one important property, namely, there exists a critical patch size for the protection zone: If the protection zone is below this size, each model behaves similarly to the non-protection zone case, but every model undergoes profound changes in dynamical behavior once the protection zone is above the critical patch size, and in such a case the endangered species is always saved from extinction. Keywords: Reaction-diffusion systems, competition model, predator-prey model, steady-state solutions, asymptotic profile.
1. Introduction It is an increasingly recognized fact that environmental changes have profound impacts on ecosystems. However, reliable scientific predictions of such impacts are extremely difficult to obtain. Obviously, mathematical modeling has a significant role to play in the study of such environmental changes. In this paper, we report some recent theoretical work (see Du-Shi 14 , DuLiang 12 and Du-Peng-Wang 13 ) that examines the effect of environmental * Research partially supported by the Australian Research Council.
50
changes in several diffusive two species population models. For each model, we consider the situation that one of the two species is endangered and a simple protection zone is created for the endangered species. The focus is on the effect of the protection zone on the dynamics of the two species systems. It is our hope that the analysis of these model systems may provide useful insights on the investigation of the real ecosystems, and induce further mathematical research along this line. To be more precise, we will consider three population models, one of competition type, two of predator-prey type. Before the introduction of a protection zone, the competition model has the form
Ut - d1llu = AU - U2
j
2
Vt - d 2 llv = J.w - V
-
CUV, x
E
0, t > 0,
duv, x
E
0, t > 0,
Ovu = Ovv = 0,
x E 00, t > 0,
u(x,O), v(x, 0) ;::: (¢) 0,
x E 0,
(1.1)
°
where d 1 , d 2 , A, J.1, c, d are positive constants, is a bounded domain in R N (N ;::: 2) with smooth boundary 00, and v is the unit outward normal of 00. This system describes the interaction of two competing species living in the spatial region 0, with population densities u and v, and dispersing rates d 1 and d 2 , respectively. The parameters A and J.1 represent the intrinsic growth rates of u and v; and c, d measure the strengths of the competition. The Neumann boundary condition means that this is a closed ecosystem. The system (1.1) is know as the diffusive Lotka-Volterra competition model, and has received extensive study in the past two decades. We are interested in the case c » d, so v is the stronger competitor and u the weaker competitor. In such a case it is well known that the semitrivial steady-state solution, (u,v) == (0,J.1), is the global attractor of (1.1). This reflects the natural biological fact that if one of the competing species is much stronger than the other, the stronger competitor will eventually sweep out the weaker competitor from the shared habitat. Before the creation of a protection zone, the first predator prey model is given by 2
Ut - d1llu = AU - u Vt - d 2 llv = J.1v - v
2
+
OvU = Ovv = 0, u(x,O), v(x,O);::: (¢)O,
buv , x E 0, t > 0, l+mu cuv , x E 0, t > 0, l+mu x E 00, t > 0, xE
0.
(1.2)
51
Here n is as before, and the coefficients are all positive. This system is known as the diffusive predator-prey model with Holling type II functional response, which has been widely used to model the interactions of a prey species U and a predator species v. We assume that the prey growth rate A is fixed and the predator growth rate JL » 1; so we have the case of a very strong predator, and it follows from standard analysis that the semitrivial steady-state, (u,v) == (O,JL), is the global attractor of (l.2). Thus in this case the prey species becomes extinct in the long run. (Note that though only the interaction of two species is considered in this model, when JL > 0, the predator species v can survive without the presence of u, implying the existence of other food sources for v.) It is widely known that the dynamical behavior of predator-prey models is very sensitive to the change in the reaction terms. A well known predatorprey model that has very different features to (l.2) is the following diffusive Leslie model:
Ut - d1t::..u = AU - u 2 - buv, x En, t > 0, Vt - d 2 t::..v = JLv - ~ , x E n, t > 0, ayu = ayv = 0, x E an, t > 0, { u(x,O), v(x,O) ;::: (:;i:) 0, x E n,
(l.3)
where n is as before and A, JL, b are positive constants. This system has the special feature that the carrying capacity for the predator v is proportional to the population size of the prey u. (Without diffusion, this model was first introduced in Leslie 24 and Leslie-Gower 25 .) As in (l.2), we assume that the prey growth rate A is fixed and the predator growth rate JL » 1; then it is expected that the positive steady-state
(u, v) ==
(~b' AJL b) ~ (0, Vb) l+JL l+JL
is the global attractor of (1.3). If such a situation happens in the real world, the prey species u would be highly vulnerable to extinction. We now describe the protection zone for the endangered species u in each of the three systems (1.1), (1.2) and (1.3). It is one of the simplest type: The v species is initially driven out from a certain subregion no of the habitat n, and then prohibited from entering no; the u species can enter and leave no freely. Thus for (1.1), no is a competition-free zone for the weak competitor u, while in (1.2) and (1.3), no is a predation-free zone for the prey species u. (One may think of a barrier along ano that blocks v but not u. If certain fishing/hunting activity of the human being is regarded as behaving similarly to a predator species, then we may think of no as a
52
fishing/hunting-free zone; in such situations, a physical barrier is usually not needed.) For simplicity of mathematical treatment, we suppose that Do is a smooth domain in RN satisfying c D. With a protection zone created this way in all three models, (1.1) is changed accordingly to
no
dlt:..u
Ut -
=
d2 t:..v =
Vt -
u2
AU -
v
/-LV -
2
-
-
> 0, x E D I , t > 0, x E aD, t > 0, x E aDo, t > 0,
c(x)uv, xED, t duv,
avu = avv = 0, avv
= 0,
(1.4)
u(x,O) ~ (¢)O,x E D; v(x,O) ~ (¢)O,x E D I ,
where DI = D \
no, and c(x)
=
CXn\oo(x)
={
n\no,
°
c when x E when x E Do.
(1.5)
Similarly (1.2) and (1.3) are respectively changed to 2
Ut -
dlt:..u =
AU -
Vt -
d2 t:..v
=
/-LV -
ovu
= ovv = 0, = 0,
avv
u 2 V
+
b(x)uv
, xED, t
> 0,
, x E DI
t > 0,
l+mu cuv
l+mu
,
x E aD, t x E aDo,
> 0, t > 0,
u(x, 0) ~ (¢) 0, xED; v(x, 0) ~ (¢) 0, x E D I
(1.6)
,
and
xED, t > 0, xED \ t > 0, x E aD,t > 0, x E aDo, t > 0, (¢) 0, x E D I ,
no,
ovv = 0, u(x, 0) ~ (¢) 0, xED; v(x, 0) ~
(1.7)
where b(x) = bXn\oo (x) = {
x n \ no, °b when when x Do. E E
(1.8)
Note that since c(x) = b(x) = 0 in Do, though V is not defined for x E Do, the interaction term in the first equation of all three systems can still be regarded as properly defined (it is identically over Do).
°
53 Since in our analysis the dispersing rates d l and d2 do not playa significant role, to simplify notations, we assume from now on that d l = d 2 = 1. We stress that this is for simplicity only, and all our results carryover to the general case. Moreover, to unify our notations, in view of (1.5) and (1.8), we will write
c(x) = 7Jb(x), with 7J» 1. With these simplifications, the three models with the same protection zone no are given by Ut -
~u =
Vt -
~v
ovu
= ovv = 0,
=
u2
AU -
v
JLV -
2
ovv = 0,
u(x,O) 2: (:;E)O,x E Ut -
~u
=
2 AU -
U
Vt-~V=JLV-V
2
-
+
ovu = ovV = 0,
u(x,O) 2: (:;E) 0, x E
-
7Jb(x)uv, x
E
n, t > 0,
nI ,
> 0, x Eon, t > 0, x E on o, t > 0, n; v(x,O) 2: (:;E)O,x E n I , duv,
x E
t
b(x)uv , x E n, t > 0, l+mu cuv ,xEnI,t>O, l+mu x Eon, t > 0,
n;
(1.9)
(1.10)
x E on o, t > 0, v(x,O) 2: (:;E) 0, x E n I ,
and Ut -
~u
Vt -
A uV
x E n, t > 0, x E n\no,t > 0, x E on,t > 0, x E on o, t > 0, V(x, 0) 2: (:;E) 0, x E nI ,
= AU - u2 - b(x)uv, v2 = MV - -;-,
ovu = ovV = 0, ovV = 0, u(x, 0) 2: (:;E) 0, x E
n;
(1.11)
with b(x) defined in (1.8). As we will see below, the effects of the protection zone on the dynamical behavior of the three systems (1.9), (1.10) and (1.11) are significantly different from each other, however, they all share an important property, namely, there exists a critical patch size for the protection zone no: If no is below this size, each model behaves similarly to the non-protection zone case, but every model undergoes profound changes in dynamical behavior once no is above the critical patch size, and in such a case the endangered species is always saved from extinction.
54
This critical patch size in all three models is determined by the same equation
Af(n o) = A.
(1.12)
Here Ap(n o) denotes the first eigenvalue of the Laplacian differential operator -~ over no under zero Dirichlet boundary conditions. It is well-known that n l C n 2 and n l -j n 2 imply Ap(nd > Ap(n 2 ). Hence the value of Ap(n o) decreases when no is enlarged. Moreover, Ap(n o) decreases to 0 as no is enlarged to the entire R N, and it increases to infinity as no is shrunk to a point. Therefore if no is a ball with radius r (denoted by B r ), then for given A > 0 we can find a unique ro > 0, depending on A, such that AP (Bra) = A. In this case Bra has the critical patch size, and any Br with r > ro (r < ro) is above (below) the critical patch size. We remark that the value of Ap(n o) also depends on the shape of no. If one keeps the volume of no fixed and only changes its shape, the value of Ap(n o) may vary, and by the classical Rayleigh-Faber-Krahn inequality (see Payne 31 ), it reaches a minimal value when no takes the shape of a ball with the given volume. Therefore, unless the shape of no is fixed (e.g., ball-shaped), equation (1.12) does not determine a unique no. To gain further insights on this critical patch size, we need the classical logistic equation over no with zero Dirichlet boundary conditions: -~w
= Aw -
w
2
in
no, w[arla = O.
(1.13)
Let us recall the well-known fact that (1.13) has a unique positive solution if and only if A> Ap(n o); we will denote this unique positive solution by w>.. Now the biological meaning of the critical patch size for the protection zone may be interpreted as follows: It is the threshold patch size of a habitat (of similar nature to n) that determines whether the prey species can survive were it the only species on that habitat surrounded by hostile boundaries. As can be seen below, for (1.9) with T} » 1, and (LlO) and (1.11) with JL » 1, the theoretical results suggest that the population distribution of the endangered species u for large time t is close to 0 outside the protection zone no, and is close to w>. inside no. In other words, u is close to W>., where W>.(x)
= {O
in
w,\(x) in
nl , no.
(1.14)
The rest of this paper is organized as follows. In section 2, we describe the results for the competition model (1.9) obtained in Du-Liang l2 . Section 3 is devoted to the predator-prey model (1.10) studied in Du-Shi l4 , and
55 section 4 considers the Leslie predator-prey model (1.11) investigated in Du-Peng-Wang 13 . We end this section with a brief account of some related research. A protection zone in a habitat, such as that in the three models considered here, destroys the otherwise homogeneous environment for the involved species. The effect of heterogeneity of the environment on ecosystems is a central problem in ecology, but very few mathematical models have been successfully used to capture such environmental effects on population dynamics. Traditionally, deterministic population models are expressed in terms of ordinary differential equations (ODEs), where spatial effects are ignored (see, for example, May29,30). To include spatial variation into consideration, reaction-diffusion systems have been widely used for such models, and through deep mathematical analysis of these systems, remarkable progress has been achieved, for example, in the understanding of ecological invasion, traveling wave, pattern formation, etc. There is, however, a limitation in almost all these reaction-diffusion models: they assume a uniform spatial environment, and most of the abstract and analytical tools that have been so successfully developed for studying them are either insensitive to whether the environment is spatially homogeneous, or they collapse once spatial uniformity is lost; a detailed explanation of this point can be found in Du-Shi 15 . Very recently, some efforts have been devoted to addressing this problem, and certain effects of heterogeneity of the environment on some competition and predator-prey models have been revealed; see, for example, Cantrell-Cosner 1 , Dancer-Du 4 , Du5- 8 , Du-Hsu lO , Du_Shi 15 ,16 , DuW ang 17 , Hutson-Lou-Mischaikow21,22, Hutson-Lou-Mischaikow-Polacik23 , LOU 26 , Lou-Martinez-Polacik27 . Some of these works were reviewed in the survey articles Du 8 and Du-Shi 15 .
2. The competition model In this section, we consider (1.9) and report the main results obtained in Du-Liang 12 .
2.1. Preliminaries We collect some preliminary results here. Linear eigenvalue problems will play an important role in our analysis. We denote by (1;,0) and (1;,0) the first eigenvalues of -~ + 1; over a bounded domain 0, with Dirichlet or Neumann boundary conditions, respectively. We usually omit 0 in the notation if 0 = D. If the potential function 1; = 0, then 1; is usually omitted
>.f
>.i'
56 from the notation. We recall some well-known properties of Af(, 0) and
Ai" (,0): (1) Af(,O) > Ai" (,0); (2) Af(I,O) > Af(2,0) if 12: 2 and 1 ¢.2, for B = D,N; (3) Af (, Or) 2: Af (, O 2) if 0 1 C 0 2.
It is easy to see that (1.9) has two semitrivial steady-state solutions
(A,O) and (0, I-t), and it follows from well-known results on the logistic equation that (A,O) attracts all the solutions of (1.9) with u(·,O) -=I- and v(·,O) == 0, and (O,I-t) attracts all the solutions of (1.9) with u(·,O) == and v(',O) -=I- 0. Moreover, standard linearization analysis shows that (A,O) is linearly stable as a steady-state solution of (1.9) if I-t < dA and it is linearly unstable if I-t > dA. To analyze the linearized stability of (0, I-t), we will need the function
° °
f(~) := Ai" (b(·)~, D).
Clearly, for any
~
f(~)
> 0, < Af(b(.)~, D) < Af(b(.)~, Do) = Af(Do).
On the other hand, for any A E (O,Af(D o )), by Theorem 2.1 in Du-Shi I5 there exists a unique ~o > such that
°
f(~o) = Ai"(b(')~o,D) = A.
,
(2.1)
These facts imply, by a standard linearization consideration, the following result.
Lemma 2.1.
(1) If A 2: Af(Do), then (0, I-t) is a linearly unstable steady state of (1.9) for any TJ > 0. (2) If < A < AP(D o), then (0, I-t) is linearly unstable if TJ < TJo := ~o/ I-t and (0, I-t) is linearly stable if TJ > TJo·
°
An important property of competition systems is that they preserve a certain order structure. Denote by K 1 , K2 the cones of all nonnegative functions in C([2") , C(Dr) respectively. Denote u ::::: u' whenever u ' - u E Kl and v ::::: v' whenever v' - v E K 2, moreover, u < u ' whenever u ::::: u' , u -=I- u' and v < v' whenever v ::::: v', v -=I- v'. Let K = Kl X K 2. It is well known that a competitive parabolic system with two species generates a monotone dynamical system (see Hess-Lazer l8 , Hsu-Smith-Waltman 2o , Smith 33 ). Our situation here is slightly different from the standard case
57
since the equations for U and v are over different spatial domains, nand n l , respectively. However, one easily checks that the standard theory carries over to our situation. Thus, we can apply the maximum principle and the theory of monotone dynamical systems (as in, for example, Matano 28 and Smith33) to obtain the following two propositions. Proposition 2.1. Suppose that (Ul' VI) and (U2' V2) are the solutions of (1.9) with continuous nonnegative initial data (¢l, ¢l) and (¢2, ¢2), respectively. Then Ul (', t) ~ U2 (-, t) and V2 (-, t) ~ VI (-, t) for any t > 0 whenever ¢l ~ ¢2 and ¢2 ~ ¢l. Moreover, if (¢l, ¢d =t (¢2, ¢2) and one of the following conditions is satisfied
=t 0, ¢l =t 0, (2) ¢2 =t 0, ¢2 =t 0, > U2(X, t) \Ix E 0 and V2(X, t) > VI (x, t) \Ix (1) ¢l
then
Ul (x,
t)
E
01
for any
t > O. Proposition 2.2. Suppose that (u,v) is the unique solution of (1.9) with nonnegative initial data (¢, ¢) E Cl(O) x Cl(Ol) satisfying, in the weak sense, -l:!.¢
~ A¢ - ¢2 - 'T/b(x)¢¢, x En,
-l:!.¢ ::::: J.l¢ - ¢2 - d¢¢, x E n l , {
ovu = 0, x
Eon,
ovv = 0, x
(2.2)
E anI.
Then u(·, t) - u(·, t/) E Kl and v(·, t/) - v(·, t) E K2 for any t' > t > O. Moreover, (U, V) = limt--->oo(u(·, t), v(·, t» exists, and (U, V) is a steadystate solution of (1.9).
Using these and the comparison principle, we can show the following result (see Du-Liang 12 for the proof): Proposition 2.3. Suppose A > Ai' ('T/J.lb(·») so that (0, J.l) is unstable. Then there exists a nonnegative steady-state solution (U, V) with U(x) > 0 \Ix EO such that any solution (u, v) of (1.9) with initial data uo, Vo =t 0 satisfies liminfu(-,t)~U, t~oo
limsupv(·,t):::::V uniformlyinx.
(2.3)
t-+oo
Furthermore, ifuo::::: U,vo ~ V, then limt--->oo(u(-,t),v(·,t» = (U, V). Suppose J.l > dA so that (A,O) is unstable. Then there exists a nonnegative steady-state solution (U ' , V') with V'(x) > 0 \Ix E 0 1 such that any solution (u,v) of (1.9) with initial data Uo,Vo =t 0 satisfies
liminfv(·,t) ~ V', limsupu(·,t)::::: U' uniformly in x. t -HX)
t-+oo
(2.4)
58
Furthermore, if Uo ~ U ' , Vo :::; V', then limt->oo(u(" t), v(-, t))
= (U' , V').
2.2. Main results
We now discuss in detail the positive steady-state solutions and the dynamical behavior of (1.9). Our analysis will be carried out according to the following four cases: (1) A> Af(n o), J.l < dA, namely (A,O) is linearly stable, (0,J.l) is linearly unstable. (2) A > Af(n o), J.l > dA, and therefore both (A,O) and (0, J.l) are linearly unstable. (3) A < Af(n o), J.l < dA, so (A,O) is linearly stable, (0,J.l) is linearly unstable if 'T/ < 'T/o and linearly stable if'T/ > 'T/o· (4) A < Af(n o), J.l > dA, and thus (A,O) is linearly unstable, (0,J.l) is linearly unstable if'T/ < 'T/o and linearly stable if'T/ > 'T/o.
Here'T/o = ~o/J.l and ~o >
2.2.1.
°is uniquely determined by (2.1).
Case 1: A > Af(n o), J.l
< dA.
As indicated above, in this case, (A, 0) is stable and (0, J.l) is unstable. Let us also note that A> Af(n o) implies that the protection zone no is above the critical patch size. We first have the following theorem on the steady-state solutions of (1.9). Theorem 2.1.
°
(1) There exists some 'T/* > such that (1.9) has no positive steady-state solution for 'T/ < 'T/*, has at least one positive steady-state solution for 'T/ = 'T/* and has at least two positive steady-state solutions for 'T/ > 'T/* . (2) Given any p E (0, J.l), there exists Tip > such that for 'T/ > Tip, (1.9)
°
has a unique positive steady-state solution (Url'VlJ) with the property VlJ(x) ~ p, 'Vx E D1 . Moreover, (UlJ' VlJ) is linearly stable and converges to (Wx,J.l) in C(D) x C(D 1 ) as'T/ ---+ 00, and if (u lJ ,vlJ ) is any other positive solution, then IlvlJlloo ---+ as fJ ---+ 00.
°
The proof of this theorem is long and technical; we refer to Du-Liang 12 for details. We note that for sufficiently large fJ (depending on p), besides (UlJ' VlJ) there exists at least one more positive steady-state solution. In order to understand the dynamical behavior of (1.9), we want to know more about these steady-state solutions. The proof of the theorem actually
59 shows that in the order induced by the cone k := (-KI) x K 2, (Ury, V,.,) is the maximal positive steady-state solution, in the sense that any other positive steady-state solution (u, v) satisfies (u, v) ~k (Ury, V,.,); i.e., u ::::: U,., in D, v ~ V,., in DI . In what follows, we provide more information on the asymptotical behavior ofthe positive steady-state solutions of (1.9) as "I ~ 00. Note that we already knew that (U,." V,.,) ~ (W'\,JL) uniformly as "I ~ 00, and if (u,."v,.,) is any other positive solution, then limry_oo Ilvrylloo = 0. If (u,." Vry) is a positive steady-state solution of (1.9), then (Ury, Wry) with w,., = TJVry is a positive solution of
{
-~u A
-tiW
ovu
=)..u - u 2
=
IIW -
t-""
-
b(x)uw, xED,
w2 TJ
duw ' x E DI ,
(2.5)
= 0, x E aD, ovW = 0, x E ODI.
This system may be viewed as a perturbation of the following problem:
{
-~u
=)..u - u 2
-~w =
-
b(x)uw, xED,
(2.6)
JLw - duw, x E Db
ovu = 0, x E
aD,
ovW = 0, x E ODI,
For later use, contrary to elsewhere in this subsection, instead of ).. > )..f(Do), we only assume).. > in the following result.
°
Proposition 2.4. Suppose).. > 0. Then pmblem (2.6) has no positive solution when JL ::::: d)", and it has at least one positive solution when JL < d)". The proof of this result uses a degree argument, which involves establishing a priori bound for all the possible positive solutions of (2.6), see Du-Liang l2 . Let S denote the set of positive solutions of (2.6). By standard elliptic regularity theory, one sees that S is precompact. It is easily checked that (0,0) and ()..,O) are nondegenerate solutions of (2.6), and hence they are isolated solutions; thus S is a compact set in C(IT) x C(ITI). We are now able to obtain a better understanding of the set of positive steady-state solutions of (1.9) for large "I.
°
Theorem 2.2. Given f > 0, there exists "Ie > such that for any "I > TJu if (Ury, v,.,) is a positive steady-state solution of (1.9) that is different fmm (U,." V,.,), then d((u,." "Iv,.,), S) < f, where d is the distance function in C(IT) x C(ITI ).
60
Now, we study the dynamics of (1.9) with TJ » 1. We have proved that for any p > 0, there is TJp such that when TJ > TJp (UT/, V1)) is the unique positive steady state of (1.9) satisfying V1) ;:::: p, and it is linearly stable. Let ~ be the attracting region of (U1)' V1)) in K, that is, A1) consists of all (uo, vo) E K such that the solution of (1.9) with initial data (uo, vo) satisfies limt->oo u(x, t) = U1)(x) uniformly for xED and limt->oo vex, t) = V1)(x) uniformly for x E D1 . Making use of the order preserving properties of competition systems, it is easy to show the following (see Du-Liang 12 ): Proposition 2.5. For any TJ > TJp, if (0, '¢o) E A1)' then any (, '¢) E K
with
°'i'- :::; 0 and '¢ ;:::: '¢O also belongs to A1).
Proposition 2.6. For TJl > TJ2 > TJp, A1)l => A1)2·
The following result shows that the attracting region A1) enlarges to the entire initial data space as TJ ----* 00 (see Du-Liang 12 for its proof). Therefore, though the semitrivial steady-state (A,O) is locally attractive in this case, but its attracting region is very small, and it is the attracting region A1) of (U1)' V1) which dominates the initial data space for large TJ. Theorem 2.3. For any (uo, vo) E K with Uo i} such that (uo, vo) E A1) whenever TJ
>
'i'-
°and Vo 'i'- 0, there is some
i}.
Remark 2.1. Combined with Proposition 2.5, we have the following stronger result: For any (uo, vo) E K with Uo 'i'- and Vo 'i'- 0, there is some i} such that (,,¢) E A1) whenever TJ > i} and (,'¢) E K, 'i'- :::; uo,'¢;:::: Vo.
°
2.2.2. Case
°
2: A > Ar(n o), J-l > dA.
In this case, we know the semi-trivial solutions (A,O) and (0, J-l) are both nondegenerate and unstable. Note that the protection zone is still above the critical patch size (A > Ar(n o». In this case, we can transform the steadystate problem of (1.9) into a fixed point problem for some compact operator A which maps the order interval [(A,O),(O,J-l)lk into itself and is order preserving. Since (A,O) and (0, J-l) are nondegenerate and linearly unstable, it is well known that A must have a stable fixed point in [(A, 0), (0, J-l)lk \ {(A, 0), (0, J-l)} (see Dancer 3 ). Since (0,0) is unstable, it follows that (1.9) has at least one stable positive steady-state solution for any TJ > 0. When TJ is large, we have the following much stronger result (see DuLiang 12 ).
61
Theorem 2.4. For sufficiently large T}, (1.9) has a unique positive steadystate solution (U'I' V1))' which converges to (W,A,p) in C(TI) x C(TId as T} ---400. Moreover, it is globally attractive, namely, if (u(x,t),v(x,t)) is a solution of (1.9) with u(x,O) 'i'- O,v(x,O) 'i'- 0, then limt--><Xlu(x,t) = U1)(x) uniformly for x E TI and limt--><Xl v(x, t) = V1)(x) uniformly for x E TIl. Thus in this case, for T} » 1, the dynamical behavior of (1.9) is clear and simple: Every positive solution converges to the unique positive steady-state
as t
---4 00.
Let us observe that in both Case 1 and Case 2, where the protection zone is above the critical patch size, when the competitor v is much stronger than u (i.e., T} » 1), the long-time population distribution of the endangered species u is close to U1) ~ W,A, which is in n1 and is W,A in no. (To be more accurate, in Case 1, we should say this is the most likely situation due to Theorem 2.3.)
°
2.2.3. Case 3: A < Ap(n o), p < dA. In this case, (A,O) is linearly stable, (0, fl) is unstable for T} < T}o and is linearly stable for T} > T}o. Moreover, we note that the protection zone no is below the critical patch size. Regarding the steady-state solutions, we have the following result.
Theorem 2.5. For T} > T}o, (1.9) has an unstable positive steady-state solution ([;1)' V1)) and V1) ---4 as T} ---4 00 in C(TIt). Moreover, for any to> 0, there exists T}, > such that if (u1)' v1)) is a positive steady-state solution of (1.9) with T} > T}" then d( (u1)' T}V1)) , S) < to, where S is the set of all positive solutions of (2.6), and d is the distance function in C(TI) x C(TI1)'
°
°
Let 81) c K be the attracting region of the semitrivial steady-state (O,p). We have the following result.
Theorem 2.6. For any (uo, vo) E K with Uo such that (uo, vo) E 81) whenever T} > ij.
'i'- 0, Vo 'i'-
0, there is some ij
As before, it can be shown that T}1 < T}2 implies B"II C B1)2' and if (uo, vo) E 8"1 then (1;, 'l/J) E 81) whenever (1;, 'l/J) E K satisfies 1; ::; uo, 'l/J 2': Vo· Thus, Theorem 2.6 implies that the attracting region B"I enlarges to the entire initial data space as T} ---4 00. (We note that for fixed T} » 1, B1) does not contain every function in the entire initial data space, since (1.9) has at least one positive steady-state.) We refer to Du-Liang 12 for the proofs of the results here.
°: ;
62 2.2.4. Case
4:
A < Ap(n o ), p,
> dA.
In this case, (A,O) is linearly unstable, (0, p,) is linearly unstable for 'Tf < 'Tfo and is linearly stable for 7] > 7]0. Clearly no is still below the critical patch size. In this case, for 7] » 1, the dynamics of (1.9) is clear and simple.
Theorem 2.7. There exists 7]* ~ 7]0 such that {1.9} has a stable positive steady-state solution when 7] E (0, 7]*), and it has no positive steady-state solution when 7] > 7]*. Moreover, when 7] > 7]*, the semitrivial steady-state solution (O,p,) attracts all the positive solutions of {1.9}. To summarize, we find that in both Case 3 and Case 4, where the protection zone is below the critical patch size, when the competitor v is much stronger than u (i.e., 7] » 1), in the long time the endangered species u becomes extinct, and thus the protection zone does not save the endangered species. (To be more accurate, in Case 3, we should say this is the most likely scenario due to Theorem 2.6.)
3. The Holling type II predator-prey model We consider (1.10) and present the main results of Du-Shi 14 in this section. A fundamental difference between (1.10) and (1.9) is that (1.10) does not have an order preserving property. This makes the understanding of (1.10) much harder. To overcome this difficulty, Du-Shi 14 makes use of the comparison principle to relate the solutions of (1.10) with the solutions of the following auxiliary equation: u t -.6.u = Au - u 2 {
-
b(x)p,
u
1 +mu
,
x E
n, t > 0,
o"u = 0,
x EOn, t > 0,
u(x,O) = uo(x) ~,'i= 0,
x E
(3.1)
n.
Such an auxiliary equation was first used in Du-Shi 16 , where system (1.10) without any protection zone but with spatially dependent coefficients (heterogeneous environment) was studied. The set of positive steady-state solutions of (1.10) can be analyzed by a standard bifurcation argument, based on abstract theory of CrandallRabinowitz 2 and Rabinowitz 32 . We fix A, c, m > 0, and take p, as the bifurcation parameter. For any p, > 0, (1.10) has two semi-trivial steady-state solutions: (A,O) and (0, p,). So we have two curves of these solutions in the space of (p" u, v) E R x C(O) x C(Ol):
ru
= {(p"A,O):
-00
< P, < oo}, rv
= {(p,,0,p,):
°< p, < oo}.
(3.2)
63 From the strong maximum principle, any non-negative steady-state solution (u,v) of (1.10) is either (0,0), or semi-trivial, or positive. The following eigenvalue problem is important for our bifurcation analysis: b..¢ + A¢ - b(x)p¢
=0
b..'¢ - p'¢ + cp¢ = 0 {
ov¢lao
= 0
ov'¢lao 1
in D, (3.3)
in Db = O.
We note that (3.3) has a solution with ¢ > 0 if and only if A
= Ai'(b(x)p,D).
(3.4)
The steady-state solutions of (1.10) are described in the following bifurcation theorem.
Theorem 3.1. (1) If A:::: AP(D o ), then (a) an unbounded continuum f 1 of positive steady-state solutions to (1.10) bifurcates fmm fu at (p,u,v) = (Pl,A,O) with /Jl := -cA/(1 + mA); (b) near (/Jl, A, 0), fl is a smooth curve (/J(s), u(s), v(s)) with s E (0,0), such that (/J(O), u(O), v(O)) = (/Jl, A, 0) and /J' (0) > 0; (c) Proj!Lfl = (/Jl,oo), and so (1.10) has at least one positive steadystate solution for any /J > /Jl, but (1.10) has no positive steadystate solution for /J ::; /J 1; (d) (O,/J) is an unstable steady state of (1.10) for any /J > 0, and there is no bifurcation of positive steady-state solutions occurring along fv' (2) If 0 < A < AP(f!O), then (a) there exists a unique /J2 = /J2(A) > 0 determined by (3.4), and a continuum f2 of positive steady-state solutions to (1.10) bifurcating fmm fv at (/J2,O,/J2); (b) /J2 (A) is strictly increasing with respect to A, and
lim /J2(A) = 0,
>.--->0+
lim
A--->[>'P(Ool]-
/J2(A)
= 00;
(c) f2 is a smooth curve near the bifurcation point (/J2,O,/J2), the bifurcation is subcritical if 0 < m < mo, and supercritical if m > mo, where mo = mO(A) is defined by
10
mo A
(3.5)
64
and
(d)
(ip1,ip2) is a positive solution of (3.3); can be extended to a bounded global continuum of positive steady-state solutions to (1.10), which meets r u at (/11, A, 0), where /11 := - 1:~'\ < o. Therefore, (1.10) has at least one positive steady-state solution for /1 E (/11, /12). Moreover, it has no positive steady-state solution if /1 ::; /11 or /1 :::: /12(1 + mA).
r2
We will see below that to understand the dynamical behavior of (1.10), the auxiliary equation (3.1) is very useful, especially for /1» 1.
3.1. Protection zone above critical size Throughout this section, we assume that Ap(n o) < A. So the protection zone no is above the critical patch size. Firstly we consider the steady-state solutions of (3.1) and use them to improve our understanding of the positive steady-state solutions of (1.10).
Proposition 3.1. Suppose that A > Ap(n o). Then
(1) For each /1 ::; 0, (3.1) has a unique positive steady-state solution Up" which is strictly decreasing in /1, and { (/1, Up,) : /1 ::; O} forms a smooth curve in R x C(IT). (2) For each /1 > 0, (3.1) has a minimal positive steady-state solution Up, and a maximal positive steady-state solution Up" and they satisfy W,\(x) < U p,(x) ::; U p,(x) < A, \Ix E
n,
(3.6)
where W,\ is defined by (1.14). (3) There exists Ji* > 0 such that for /1 > Ji* , Up, = Up" and (3.1) has a unique positive steady-state solution, which we denote by Up" and {(/1, Up,) : /1 > Ji*} is a smooth curve. Moreover, as /1---> 00, Up, ---> W,\ in C(IT). Making use of Propositions 3.1, we can now have a better characterization of the set of positive steady-state solutions of (1.10), especially when /1 is large.
Theorem 3.2. Suppose that A > AP(OO). Then
(1) For any /1 > /11, if (u,v) is a positive steady-state solution of (1.10), then Up,+clm(x)::; u(x)::; Up,(x), max{/1,O}::; v(x)::; /1 + c/m,
(3.7)
65 where Up, and Up, are the minimal and maximal steady-state solutions of (3.1), respectively. (2) There exists p,* > such that (1.10) has a unique positive steady-state solution (up"vp,) when p, ~ p,*, and (up"vp,) is linearly stable in the sense that Re(TJ) > if TJ is an eigenvalue of the linearized eigenvalue problem at (up" vp,). Moreover, when p, ~ 00, up, ~ W>. uniformly in Il, and vp, - p, ~ uniformly in Ill.
° ° °
Next we examine the long-time dynamical behavior. Again we will make use of (3.1). Recall that, by Proposition 3.1, (3.1) has a unique positive steady-state solution Up, if p, t/. (0, Jl*), and for each p, E (0, Jl*), it has a maximal positive steady-state solution Up, and a minimal positive steadystate solution U w The following result shows that the dynamics of (3.1) is largely determined by these steady-state solutions. Proposition 3.2. Suppose A > (3.1).
Af(n o)
°
(1) If p, ~ or p, ~ Jl*, then u(x, t) t ~ 00. (2) IfO
and let u(x, t) be a solution of
~ Up,(x) uniformly for x E
U p,(x) ~ limt~oou(x, t) ~ limt~oou(x, t) ~ U p,(x), uniformly for x E
n
as
(3.8)
n.
Using Proposition 3.2 and standard comparison argument, it is not hard to derive the following results on the asymptotic behavior of (1.10). Theorem 3.3. Suppose A of (1.10).
> Af(n o)
and let (u(x, t), v(x, t)) be a solution
+ mA),
then limt~oo u(x, t) = A uniformly for x E and limt~oo v(x, t) = uniformly for x E Ill. (2) If p, > -cA!(l + mA), then
(1) If p, < -d/(l
°
Il,
U p,+c/m(x) ~ limt~oou(x, t) ~ limt ..... oou(x, t) ~ U p,(x) uniformly in Il, -c max{p" O} ~ limhoov(x, t) ~ limt~oov(x, t) ~ p, + - uniformly in nl · m
Theorem 3.3 shows that for any p, > P,l = -cAI(l + mA), the order interval [Up,+c/m' Up,] x [max{p"O},p, + clm] c C(Il) x C(Il l ) attracts all the positive solutions of (1.10). By Theorem 3.2, (1.10) has at least one positive steady state solution in the attracting region, but it is unclear
66
whether (1.10) can have multiple steady state solutions or even periodic solutions in the attracting region. However when J.L is large, we have shown in Theorem 3.2 that (1.10) has a unique locally asymptotically stable positive steady state solution (uJ.L,vJ.L). Our next result shows that (uJ.L,vJ.L) is actually globally asymptotically stable. Theorem 3.4. Suppose that A > Af(!lo). Then there exists J.L* > 0 such that if J.L ~ J.L*, and if (u(x, t), vex, t)) is a solution of (1.10), then
limt-><Xl u(x, t) = uJ.L(x) and limt-><Xl vex, t) = vJ.L(x) uniformly for x and x E !l1, respectively.
E
!l
This result shows that when the predator is strong (11 ~ 1), the endangered prey species can be saved from extinction if the protection zone is above the critical patch size. Let us also recall that for 11 ~ 1, uJ.L is close to W>., which is zero in !l1 and is equal to w>. > 0 in !lo, while vJ.L is close to J.L.
3.2. Protection zone below critical size The results in Du-Shi 14 show that, when the protection zone !lo is below the critical patch size, namely Af(!lo) > A, then the behavior of (1.10) is similar to the non-protection zone case studied in Du-Shi 16 . In particular, the prey species will become extinct when the predator is strong (J.L ~ 1). Again we will use (3.1) to help with the analysis. The positive steadystate solutions of (3.1) are described in the following result. Proposition 3.3. Suppose that 0 < A < Af(!lo), and 111,112 are defined as in Theorem 3.1. Then there exists fj,* E [J.L2,00) such that (3.1) has no
positive steady-state solution if J.L > fl*, and it has at least one positive steady-state solution for J.L < fl* . Moreover, (1) when J.L :::; 0, (3.1) has a unique positive steady-state solution UJ.L(x), and {(J.L,UJ.L): J.L:::; O} is a smooth curve; (2) for 11 E (O,fj,*), (3.1) has a maximal positive steady-state solution U J.L(x), and U J.L is strictly decreasing with respect to 11; (3) for 11 E (0,J.L2), (3.1) has a minimal positive steady-state solution U J.L(x) , and U J.L is strictly decreasing with respect to J.L; (4) if fl* > J.L2, then (3.1) has a maximal positive steady-state solution for J.L = fj, *, and has at least two positive solutions for J.L E (J.L2, fl*); (5) a sufficient condition for fl* > J.L2 is m > mo with mo given by (3.5) with (P2 = 0, and all these solutions can be chosen from a continuum of positive solutions bifurcating from (J.L2, 0).
67 With the help of Proposition 3.3, we can gain a good understanding of the long-time dynamical behavior of (3.1), which can then be used to prove the following result on the dynamical behavior of the full system (1.10).
Theorem 3.5. Suppose that 0 < A < Ap(n o). Then all solutions (u(x, t), v(x, t)) of (1.10) are globally bounded, and v(x, t) satisfies max{tL,O} ::::: limt->oov(x, t) ::::: limt->oov(x, t) ::::: max{p, + ~,O}. m Moreover, the following conclusions hold:
(1) If p, <
P,l, then limt->oo u(x, t) limt->oo v(x, t) = 0 uniformly for x (2) If p, > f1*, then limt->oo u(x, t) limt->oo v(x, t) = p, uniformly for x (3) If P,l < P, < fl* - elm, then
= A uniformly for x DI .
E nand
E
= E
0 uniformly for xED and
DI .
U l1+ c / m (X) ::::: limt->oou(x, t) ::::: limt->oou(x, t) :::::
ul1(x)
uniformly in D,
max{p"O} ::::: limt->oov(x, t) ::::: limt->oov(x, t) ::::: p, + ~ uniformly in m
DI .
We note that when p, » 1, u(x, t) -> 0 as t -> 00 uniformly for xED, and thus the endangered prey species becomes extinct when the protection zone is below the critical patch size.
4. The Leslie predator-prey model In this section, we introduce the work of Du-Peng-Wang l3 on the Leslie predator-prey model (1.11). The dynamics of the ODE counterpart of (1.11) was investigated in Hsu-Hwang l9 , and it was shown that the unique positive equilibrium
(u * ,v *) ==
(A AP,) 1 + p,b' 1 + p,b
is the global attractor of the system. In Du-Hsu lO , the non-protection case of (1.11) was considered, and the same constant positive equilibrium (u*, v*) was shown to be the global attractor of (1.11) (with no = 0), at least when p,b ::::: 4. (It was conjectured in Du-Hsu lO that this restriction on p,b is unnecessary, but the conjecture has not been proved so far.) So only simple dynamics is expected for this system in the non-protection zone case. Let us note the great difference between the two predator-prey models in the non-protection zone case: The Leslie predator-prey model always has a
68
positive steady-state (u*, v*) but the Holling type II predator-prey model (1.10) (with no = 0) has no positive steady-state when JL » 1. From now on, we only consider (1.11) with no -I- 0. By a standard degree argument one proves that (1.11) has a positive steady-state solution (u,v) for all .x, JL, b > 0: Theorem 4.1. For any tive steady-state solution.
.x, JL, b > 0,
problem (1.11) has at least one posi-
We believe that (1.11) has aunique positive steady-state solution and it is the global attractor, but a proof for such a conclusion seems still beyond the reach of current techniques. Instead, we will concentrate on the analysis of the steady-state solutions for JL » 1. It turns out that the behavior of the positive steady-state solutions of (1.11) is determined by several limiting problems. The first of these is the degenerate logistic equation over n with Neumann boundary conditions: -!:l.Z
=
.xZ - b(X)Z2 in
n,
ollZlan
=
o.
( 4.1)
It is known (see, for example, Du-Huang l l ) that the degenerate logistic equation (4.1) has a unique positive solution if and only if 0 < .x < .xp(n o), which we denote as Z)... The second of these limiting equations is the following boundary blowup problem:
{
-!:l.V ollV
V =
= .xV =0 00
bV 2 in
on on
n \ no, an,
(4.2)
ono .
By Du-Huang l l , the boundary blow-up problem (4.2) has a unique positive solution for any .x E R 1 , which we denote by V)... The third limiting problem is the classical logistic equation (1.13), which has a unique positive solution w).. when .x > .xp(n o), and has no positive solution otherwise. We are now able to describe the main results on the behavior of the positive steady-state solutions of (1.11) for JL » 1. Theorem 4.2. Assume that.x < .xp(n o). Then, there exists a large JL* depending only on.x, b, n and no such that when JL > JL*, (1.11) has a unique positive steady-state solution (uJ.t,vJ.t); moreover, (uJ.t,vJ.t) is linearly stable, and when JL -; 00, JLuJ.t -; Z).. uniformly on nand vJ.t -; Z).. uniformly on
n
1.
69
Theorem 4.3. Assume that A> Af(Oo) and let J.L ~ 00; then for every positive steady-state solution (uJ.L' vJ.L) of (1.11), the following hold: (i) uJ.L ~ W>. uniformly on n, where W>. is given by (1.14); (ii) J.LuJ.L ~ vA, vJ.L ~ V>. uniformly on any compact subset ofn\
no.
lt is interesting to note that the functions Z>., V>' and the quantity Af(Oo) arising from our investigation of (1.11) are also intrinsically related in a different context (see Du-Huang l l ): If z(t, x) is an arbitrary positive solution of the corresponding parabolic equation of (4.1), then
t~~z(t,x) =
o
if A:::; 0,
Z>.(x)
if 0 < A < Af(Oo),
{ v>' (x)
if A ~ Af(Oo),
where
v>'(x)
= {ooV>'(X) for x
E
for x E
n \ no, no.
We now consider the biological implications of the above mathematical results. Compared with the original non-protection zone system, in the case that the protection zone is below the critical patch size, we find from Theorem 4.2 above that the prey species similarly has very low population level, and the predator species has population level close to a certain x-dependent positive function (Z>.) independent of the value of J.L. The function Z>. is finite but is easily shown to be greater than Alb, the approximate population level of the predator in the non-protection zone case. In sharp contrast, when the protection zone is above the critical patch size, the conclusions in Theorem 4.3 above reveal significantly different behavior of the two species: The population distribution of the prey species is no longer uniformly small over the entire habitat, instead, it is close to a fixed x-dependent positive function (w>.) inside the protection zone; the predator population also exhibits a sharp change, it is close to an x-dependent function (V>') which is unbounded near the boundary of the protection zone. (We can also easily show that V>'(x) > Alb on Next we compare (1.11) with (1.10). When the protection zone is below the critical patch size (Af(Oo) > A), for large J.L, the qualitative behavior of both models is similar to the case that no protection zone is introduced. If the protection zone is above the critical patch size (Af(Oo) < A), for large J.L, (1.10) has a unique positive steady-state solution (ii, v), with ii close to 0 over 0 \ and close to w>. given by (1.13) in 0 0 , and v ~ J.L over 0 \ So v is uniformly large in its restricted habitat. In contrast,
n\ no.)
no.
no
70
when >.p(no) < >., for large J-L, any positive steady-state solution (u,v) of (1.11) satisfies v ~ vA, where VA is a boundary blow-up solution given by (4.2), while u behaves like u. Thus, for (1.11), the population distribution of the predator is far from uniform in its restricted habitat, and is largely independent of J-L. To explain this phenomenon, we need to look at the fine behavior of u over n \ no, namely J-Lu(x) is close to VA(x) in this region. Thus although u(x) is small over n \ no, but its value near ano is far bigger than that away from no. The assumption that the carrying capacity of the predator in (1.11) is proportional to the prey population suggests that the predator is very sensitive to the behavior of the prey population, which seems to be the biological reason responsible for the very uneven distribution of the predator population over n\no, and for its independence of J-L. The mathematical analysis for (1.11) is very different to that for (1.10), and it is much more difficult to handle. It relies, in particular, on various blowing up arguments and careful elliptic estimates. A key step in the mathematical analysis is the following result: Proposition 4.1. Let (ui-" vi-') be a positive steady-state solution of (1.11).
Then lim (Ui-" Vi-') = (0,0) in C(n) x C(n 1 ), i-'--->OO J-L
and
We refer to Du-Peng-Wang 13 for more details. References 1. R.S. Cantrell and C. Cosner, Spatial Ecology VIa Reaction-diffusion Equations, John Wiley & Sons Ltd, 2003. 2. M.G. Crandall and P.B. Rabinowitz, Bifurcation from simple eigenvalues,
J. F\mc. Anal., 8 (1971), 321-340. 3. E.N. Dancer, Upper and lower stability and index theory for positive map-
pings and applications, Nonlinear Analysis, 17 (1991), 205-217. 4. E.N. Dancer and Y. Du, Effects of certain degeneracies in the predator-
prey model, SIAM J. Math. Anal., 34 (2002), 292-314.
71
5. Y. Du, Effects of a degeneracy in the competition model, part I. classical and generalized steady-state solutions, J. Diff. Eqns., 181(2002), 92-132.
6. Y. Du, Effects of a degeneracy in the competition model, part II. perturbation and dynamical behaviour, J. Diff. Eqns., 181(2002), 133-164.
7. Y. Du, Realization of prescribed patterns in the competition model, J. Diff. Eqns., 193(2003), 147-179.
8. Y. Du, Spatial patterns for population models in a heterogeneous environment, Taiwanese J. Math., 8(2004), 155-182. 9. Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol.1 Maximum principles and applications, World Scientific, New Jersey, 2006. 10. Y. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203(2004), 331-364. 11. Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic
and parabolic equations, SIAM J. Math. Anal., 31(1999), 1-18. 12. Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Eqns., 244(2008), 61-86. 13. Y. Du, R. Peng and M.X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, preprint, Univ. of New England, 2008. 14. Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Eqns., 229(2006), 63-9l. 15. Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in Nonlinear Dynamics and Evolution Equations, Eds. Hermann Brunner, Xiao-Qiang Zhao and Xingfu Zou, Fields Institute Communications Vol. 48, American Math. Soc., 2006, pp95-135. 16. Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359(2007), 4557-4593. 17. Y. Du and M.X. Wang, Asymptotic behavior of positive steady-states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136(2006), 759-778. 18. P. Hess and A.C. Lazer, On an abstract competition model and applica-
72 tions, Nonlinear Anal., 16 (1991), no. 11, 917-940. 19. S.B. Hsu and T.W. Hwang, Global stability for a class of predator-prey
systems, SIAM J. Appl. Math., 55(1995), 763-783. 20. S.B. Hsu, H.L. Smith and P. Waltman, Competitive exclusion and coex-
istence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. 21. V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources
versus Lotka- Volterra dynamics, J. Diff. Eqns. 185(2002), 97-136. 22. V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition mod-
els with small diffusion coefficients, J. Differential Equations 211(2005), no. 1, 135-161. 23. V. Hutson, Y. Lou, K. Mischaikow and P. Polacik, Competing species
near a degenerate limit, SIAM J. Math. Anal. 35(2003),453-491. 24. P.H. Leslie, Some further notes on the use of matrices in population math-
ematics, Biometrica, 35(1948), 213-245. 25. P.H. Leslie and J.C. Gower, The properties of a stochastic model for
the predator-prey type of interaction between two species, Biometrica, 47(1960), 219-234. 26. Y. Lou, On the effects of migration and spatial heterogeneity on single
and multiple species, J. Differential Equations 223(2006), 400-426. 27. Y. Lou, S. Martinez and P. Polacik, Loops and branches of coexistence
states in a Lotka- Volterra competition model, J. Differential Equations 230(2006), 720-742. 28. H. Matano, Existence of nontrivial unstable sets for equilibriums of
strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, Sect. lA, Math., 30(1984),645-673. 29. R.M. May, Stability and Complexity in Model Ecosystems, Princeton
Univ. Press, Princeton, 1973. 30. R.M. May, Theoretical Ecology: Principles and Applications, Second Edi-
tion, Blackwell Scientific Publishings, Oxford, 1981. 31. L.E. Payne, /soperimetric inequalities and their applications, SIAM Rev. 9(1967), 453-488.
73 32. P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems,
J. Func. Anal., 7(1971), 487-513. 33. H.L. Smith, Monotone Dynamical Systems, Math. Surveys and Monographs, Vol. 41, Amer. Math. Soc., 1995.
74
THE STATE OF THE ART FOR A CONJECTURE OF DE GIORGI AND RELATED PROBLEMS Alberto Farina
LAMFA - CNRS UMR 6140 UniversiU de Picardie Jules Verne Faculte de Mathematiques et d'Informatique 33, rue Saint-Leu 80039 Amiens CEDEX 1, France • E-mail: [email protected] Enrico Valdinoci
Universita di Roma Tor Vergata Dipartimento di Matematica via della ricerca scientifica, 1 00133 Rome, Italy • E-mail: [email protected]
1. De Giorgi conjecture
In Ref. DG79, the following striking question has been posed: Conjecture 1.1. Let
U
E C2(]Rn, [-1, 1]) satisfy
!1u + U
-
u3
= 0 and
GnU>
(1)
0
in the whole ]Rn. Is it true that all the level sets of U are hyperplane, at least if n
~
8?
The purpose of this note is to discuss the state of the art for such a problem and present some related results. Given a function v : ]Rn -+ ]R and kEN, with k ~ n, we say that v is k-dimensional (or, for short, kD) if there exists a function Va : ]Rk -+ ]R in such a way that, after a suitable rotation of the space, V(Xl" .. ,xn ) = Va(Xl' ... ,Xk). That is, v is said to be kD if it depends only on k variables, up to rotations. It is easily seen that the claim of Conjecture 1.1 that the level sets of u are hyperplanes is equivalent to u being 1D.
75 1.1. (Crude) physical motivation
The PDE in (1) has important physical applications, since it arises in the theory of superconductors and superfluids GP58 ,Lan67 and in the study of interfaces in both gasses and solids.Row79,Ac79 It also have some cosmological implications. Car95,GT99 Let us briefly discuss a (very raw indeed) physical derivation of the model. We are given some substance in some container, called, say, 0, which may exhibit two phases, which we label with "-1" and "+1", and we would like to describe mathematically the pattern and the separation of such phases. Our ansatz could be that the interface formation is driven by a variational principle, that is the pattern is the outcome of the minimization of some kind of energy. For this, we may consider conceivable that the energy density is just some "double well" function W such that W(±l) = 0 and W(r) > 0 if r i= ±l. This would lead the interface problem to be run by the minimization of an energy of the form
Q:o(u; 0) =
L
W(u(x)) dx
where the function u(x) represent the states of the substance at the point x E O. One quickly realizes that this cannot be a satisfactory model, since any function attaining only the values -1 and + 1 minimize the energy Q: o (and, in fact, it make it vanish). In particular, the separation between the two phases could be as wild as possible and the energy would not be affected! Since, of course, in any physical application, there will be some kind of friction, or inertia, preventing sudden and unmotivated phase changes, one needs to add to Q: o a term that "penalizes" the formation of unnecessary interfaces. This may be accomplished by adding to Q: o a small gradient term, that is by looking at the energy
Q:(u;O) =
r ':'[\7U(X)[2 + ~W(u(x))dx,
Jo 2
f
where f > 0 is a small parameter. Such small gradient term indeed cuts the interfaces as much as possible, in the sense that the minimizers of Q: turn out to be smooth functions interpolating -1 and + 1 whose level sets approach hypersurfaces of least possible area.Mod87,CC95b Note that, in general, the pure phases ±1 are not attained by the minimizers of Q:, thus we should rather expect a "sharp" transition between
76
states close to -1 to states close to +1. Up to a space dilatation, we may focus on the case E := 1. The PDE in (1) is then obtained by taking the explicit potential W(r) := (1- r2)2 /4. In this physical interpretation, Conjecture 1.1 states that, at least in low dimension, global phase transitions have a flat interface, or, by a blow-up argument, that the phase, locally, just depends on the distance from the interface. Of course, the model has been obtained quite sloppily. Some modifications may be made to keep track of nonelastic drags of gradient-type and of the effect of sticky boundaries. In particular, following Ref. Lad67, one may relate anelasticity to p-Laplacian-type operators and consider energies of the type
~p(u; n) = [
.!.1V'u(x)[P + W(u(x)) dx.
lop
(2)
Boundary effects have been dealt with in Refs. ABS98,CSM05 and they are related to fractional operatorsSV07 via some results of Ref. CS07. Similar models arise in combustion and fluid dynamics: AC81 in this case, the energy takes the form of
~*(u;n) =
in ~1V'u(x)[P +
X(-l,l)(U(X)) dx,
(3)
where X is the characteristic function. In particular, u in this case represents the stream function of an ideal fluid. The boundary of the fluid corresponds to the free surface {lui = I}, where the velocity of the fluid is balanced by the pressure via Bernoulli's law. Related problems also arise for fully nonlinear elliptic operatorsFar07,DSS08 and in the Heisenberg group.BP02,BL03
1.2. Possible motivation for the conjecture Let us now give a very heuristic argument to justify Conjecture 1.1. Let u be as requested in Conjecture 1.1, take E > 0 and let
(4) Now, the monotonicity assumption in (1) seems to suggest that the phase transition happens in a straight, minimal way. Then, by Refs. MM77, Mod87,CC95b, the level sets of u£ are close to a surface of minimal area (the monotonicity assumption in (1) in fact suggests that this is a global minimal graph, namely it is the graph of a solution of the minimal surface equation in jRn-l).
77
Since global minimal graphs are flat for n - 1 :::; 7, due to Bernsteintype Theorems, Giu84 it follows that the level sets of u€ are close to a flat hyperplane. Now, since elliptic problems are somehow "rigid", we may suspect that once {u€ = c} is close enough to a hyperplane, it is a hyperplane itself (for instance, in analogy with the fact that bounded harmonic functions are constant). By scaling back, this would give that {u = c} is a hyperplane. Then, the level sets of u would be parallel hyperplanes and thus u would be ID, as asked by Conjecture 1.1! Of course, several gaps need to be filled in the above argument. First of all, no minimality condition is explicitly required by Condition 1.1, so the results of Refs. Mod87,CC95b are not directly applicable. Also, one would need to proof the rigidity argument. Finally, the monotonicity condition in (1) does not assure, in principle, that the level sets of u are complete graphs (a counterexample possibly being
U(Xl,'" ,XN) = ,(Xl
+e
XN ),
(5)
for a monotone and bounded function ,). 2. Available results As we will see, Conjecture 1.1 is completely settled for n = 2,3. Remarkably, in such cases, the ID symmetry of the solution is true for any semilinear equation (once the nonlinearity is sufficiently smooth: in particular, there is no need to require it to be a double well potential). When n 2: 4, the conjecture is open and no counterexample is available (not even for nonlinearities different from double well potentials, though important contributions in this direction have been given in Ref. JM04). * However, several deep results have recently appeared also for 4 :::; n :::; 8, as we will discuss below. We consider the following notation. Given a solution of the equation
/).u
+ f(u) = 0
(6)
* Added in proof. When this review was completed, Ref dPKW08 appeared. There, the authors construct an example of a minimal solution u E C 2(JR9), satisfying ~u+u-u3 = o and 09U > 0 in JR9, with lim U(X',X9) = ±1, xg--+±<x>
and such that {u = O} is at a bounded distance from the saddle minimal surface of Ref BDGG69. Particularly, such u is not 1D. This implies that the assumption n ::::: 8 in Conjecture 1.1 cannot be removed.
78
in the whole JRn, with f locally Lipschitz, we say that u is stable if
(7) for any smooth and compactly supported function ¢: JRn ----> JR. The set gu in (7) is simply the set in which J'(u(x)) is defined (such complication is needed since f may not admit derivative everywhere), namely one can consider the sets
9
:= {t
E JR s.t. f'(t) exists},
N:=JR\9,
Nu := u-i(N) = {x E JRN s.t. u(x) EN}, and
gu:= JRN \ Nu
in order to give sense of the last integrand in (7). Of course, for smooth f, the set gu is the whole JRn. It is well known that if an u > 0 in JRn, then u is stable: we recall here the proof of this fact (we take f to be smooth for this computation, for a general result see Section 7 in Ref. FSV08). Differentiating (6), we have
Integrating by parts against ¢2 / an u, we obtain 0=
r 2¢'l¢·
'l(anu) anu
J'Rn
: ; 1"
['l¢[2 - J'(u)¢2 dx,
where the Cauchy-Schwarz inequality has been exploited, and this yields (7).
2.1. The case n = 2 Conjecture 1.1 holds for n = 2, as first proved in Ref. GG98 (see also Ref. BCN97). In fact, Conjecture 1.1 holds for any Ci-nonlinearity, as shown in Theorem 1.1 of Ref. GG98, and, furthermore, according to Ref. FSV08, the following more general result holds true:
Theorem 2.1. Let f be locally Lipschitz, and let u E C 2 (JR 2 ), with ['lu[ E LOO(JR 2 ), be a stable solution of (6) in the whole JR 2. Then, u is 1D.
79
When the stability condition is removed from Theorem 2.1, then u does not need to be 1D. For instance, Ref. DFP92 constructed saddle solutions that are not stable (and, in particular, not monotone). A detailed spectral analysis of such unstable solution was performed in Ref. Sch95. See also Ref. Gui08 for further properties of saddle solutions. We call attention to the fact that the extension to the locally Lipschitz nonlinearities, which has been performed in Ref. FSV08, is not of merely academic interest, since several physical applications are run by locally Lipschitz forces. For instance, take a series of close-by particles (say, of negligible mass) each joined to the nearest ones horizontally by an elastic spring and hanged vertically to a suspension bridge and let u the vertical displacement of a particle (an analogous model would be a suspended elastic membrane).
Fig. 1.
Model of a suspension bridge
Then, the tension of the horizontal springs is, as standard, approximated by L:1u and it is balanced by the tension of the ropes of the bridge. When
80
one particle moves downwards, the rope tries to pull it back up: this force may be assumed, for small displacements, to be elastic (say, equal to u, if we neglect Hook constants). On the converse, if the particle is pulled up, the rope bends up and it does not make a stand. Thus, the equilibrium configurations of the suspension bridge model are driven by equations of the type ~u(X)
+ u(x)X(-oo,O)(u(x))
= 0,
and the latter is just a locally Lipschitz nonlinearity. We remark that Theorem 2.1, besides the extension to Lipschitz nonlinearities, provides the further generalizations with respect to the problem in Conjecture 1.1: • the assumption of monotonicity is weakened to stability, • the solution is not necessarily bounded (only a bound on the gradient is needed: for instance linear functions like u(x) = Xn are unbounded monotone solutions of (6) with bounded gradient), • more singular and degenerate operators than the one in (6) may be taken into account (see (14) below). Let us give two quick proofs of Conjecture 1.1 when n = 2, which are indeed valid for solutions of (6) (assuming here, for simplicity, that f E
C 1(JR)).
2.1.1. First proof of Conjecture 1.1 for n = 2 We follow here the version by Ref. Far03 of the proof by Ref. GG98 (see also Ref. BCN97). With a slight misuse of notation, we identify vectors in JR2 with complex numbers: thus we write \1u in complex polar coordinates as \1u =: z = pe iO E C.
Then, by differentiating (6), we obtain izing eiO ,
~z
+ f'(u)z
= 0 and so, by factor-
~p - pl\1012 + f'(u)p + i (p~O + 2\1 p. \10)
= O.
We now take the imaginary part of such equation (note that this choice completely kills the nonlinearity f, which only appears in the real part!) and we conclude that div (p2\10) = O.
(8)
81
We now observe that 0 < IV'UI = p is bounded by standard elliptic est imates. GT01 Also, the monotonicity assumption in (1) implies that V'u does not "turn backwards" , that is () E [0,7TJ, and so, in particular, () is bounded. Following Ref. MM80, it is now quite tempting to search for a Liouvilletype Theorem for equation (8). If a result of this kind held, it would give that () is constant, say () = ()o, and so the level sets of U would be straight lines parallel to {() = () 0 + 7T /2}, yielding that U is 1D. Indeed, as pointed out by Ref. BCN97, such Liouville-type Theorem indeed holds: the proof is not hard and it is based on the integration of (8) against a cut-off function T E C(f(B2R)' with T = 1 in B R, taking R arbitrarily large. To make such computation work, one needs the fact that
r
p2()2 dx :::; constR2
(9)
JB2R
for large R, which is obviously true here since we are in JR2 and so the Lebesgue measure of B2R is of order R2. See also Ref. GG98, where the spectral properties for the linearized equation of (6) are studied in connection with Ekeland Principle. 2.1.2. Second proof of Conjecture 1.1 for n = 2 We follow here an idea of Ref. Far02 developed in Ref. FSV08 for a very general setting, which is based on a weighted Poincare-type formula found in Refs. SZ98a,SZ98b. Given any smooth compactly supported function
L"
/V'IV'ul/2
+ ~V'
+ lV'uI21V'
V'1V'uI 2 - J'(u)lV'uI2
Also, from (6),
L"
J'(u)lV'uI2
=-
t1 j=l
=
t, L"
D..UjUj
=
R"
r ~ IV'UjI2
8 j (J(u»)Uj
t1 j=l
dx
IV'UjI2
+ Uj V'Uj
. V'
R"
+~ 2
r V'1V'uI 2 . V'
dx.
(11)
82
By inserting (11) in (10), we conclude that
{ Jrvu-l-O}
(tIV'IV'UjI12 -IV'IV'uln
From a differential geometry formula,sz98a,sz98b one writes such inequality as
{
(IV'uI 2e:2+ IV'TIV'uln
(12)
J{''V'u#-O}
Here above, we denoted by V'T the tangential gradient along the level set of U and by e: 2 the sum of the squares of the principal curvatures of such a level set (note that the level set of U is a smooth manifold on {V'u =I- O}). Formula (12) may be seen as a weighted Poincare-type inequality, since it relates the weighted L 2 -norm of any test function with a weighted L2_ norm of its gradient. Remarkably, the weights are important geometric objects coming from a stable solution of (6). We now perform a capacity argument, by fixing R > 0 and taking
~
log
{
~o~l~g Ixl if
o
if Ixl ::; JR, JR < Ixl < R, if
Ixl ::::
R.
Such choice, by taking R arbitrarily large, yields that e: and V'TIV'ul vanish identically. Therefore, the level sets of U are parallel flat hyperplanes, thence U is 1D. We observe that this proof can be carried out also for locally Lipschitz nonlinearities (in particular, Theorem 2.1 may be proved this way) and, as we will see, it can be adapted to other situations too.
2.2. The case n = 3 In analogy with Theorem 2.1, we have:
Theorem 2.2. Let f be locally Lipschitz, and let U E C 2 (JR 3 ) n U)O(JR 3 ) be a solution of (6) in the whole JR3 such that fhu > O. Then, u is 1D. Theorem 2.2 was proved in Ref. AAC01 under the assumption that f E C 1 (JR), by generalizing the work in Ref. ACOO which was done for a more restricted class of nonlinearities. The case of locally Lipschitz nonlinearities has been dealt with in Ref. FSV08. We observe that the approach in Refs. ACOO,AAC01 cannot deal with the case in which l' is not continuous, due to a limit argument,
83
therefore Ref. FSV08 utilizes a geometric weighted Poincare-type formula, as in Refs. SZ98a,SZ98b, which better behaves with respect to the limit. Hence, though with several complications, the proofs in Section 2.1.1 and 2.1.2 may be adapted to the case n = 3. Indeed, the proof in Section 2.1.1 seems to depend on the complex notation, but this may be avoided simply by looking at the vector (OXlujoXnU, ... ,OXn_lujoXnU): this way, the computations needed to obtain (8) may be carried out with suitable modifications. But, in a somewhat more fundamental way, the proof in Section 2.1.1 relies on estimate (9), in order to obtain the Liouville-type Theorem (see also Ref. Far07 for further comments on this point). But, as a matter of fact, by a profile analysis, it is shown in Ref. AAC01 that
r IV'ul dx :S const R2 2
iBR
(13)
for any ball B C ]R3, which implies (9). This makes the proof in Section 2.1.1 work also for the case n = 3. The proof in Section 2.1.2 may also be adapted to the case n = 3: the idea is here to perform a capacity argument on the graph of u, which "behaves like ]R2", being a hypersurface in ]R3. We refer to Refs. Far02,FSV08 for full details. In Ref. FSV08, an argument alternative to the Liouville-type Theorem is also given. We also recall that a very general result is given in Ref. FSV08. Namely, Theorem 2.2 is proved there for (possibly degenerate or singular) elliptic equations of the form div (a(IV'uI)V'u)
+ f(u) = o.
(14)
In particular, Ref. FSV08 comprises the p-Laplacian operator, for a(t) = 2 , and the mean curvature one, for a(t) = 1jyT+i2. For related lower dimensional symmetry results see also Refs. DG02,Far03. As a byproduct, the technique of Ref. FSV08 also gives a general Bernstein-type Theorem for n = 2, thus stressing the link between general phase transition problems and suitable versions of minimal surfaces. We remark that estimates analogous to (13) cannot hold in higher dimension, since, as can be easily checked, even for 1D solutions one cannot expect anything better than
tp -
r IV'ul iBR
2
dx :S const R n -
1
.
This suggests that profoundly new ideas are needed to treat the case n 2: 4.
84
2.3. The case 4
~
~
n
8
Conjecture 1.1 is still open when n :::: 4, but some very important progress has been recently made after Refs. Sav03,Sav08: Theorem 2.3. Let u be as requested in Conjecture 1.1 and let n < 8. Assume, furthermore that lim X n -4+OO
and
lim Xn -4-00
u(x',x n ) = 1 u(x',x n )
=
-1.
(15)
Then, u is 1D. We observe that Theorem 2.3 settles Conjecture 1.1 when n :::; 8 under the additional condition in (15), namely that the "profiles" of the solution at ±oo are ±1 (see also (18) below). Though (15) was not originally present in Conjecture 1.1, such additional assumption is natural and coherent with the phase transition setting. Also, it is compatible with the problem (indeed, if Conjecture 1.1 holds true, then, a posteriori (15) is also true). On the other hand, when (15) holds and the limits are attained uniformly, the ID symmetry always holds. Indeed, the following result, conjectured by G. W. Gibbons,Car95,GT99 is true in any dimension, under a uniform limit assumption: Theorem 2.4. Let u Suppose
E
C 2 (lRn) n Loo(JR.n) satisfy (1) in the whole JR.n.
that (15) holds and that the limits are attained uniformly w.r.t. x'
(16)
Then, u is ID. Note that no monotonicity requirement on u is needed for Theorem 2.4. Theorem 2.4 was first proven independently and with different methods by Refs. Far99,BBGOO,BHMOO. In Ref. FV08a it is also shown that the uniform control of only one profile is enough to obtain that u is ID in case u is a minimal solution. We observe that (16) carries important geometric information, since it gives a priori that the monotonicity direction is the nth variable: this is not the case of condition (15), since a ID solution in the nth direction may be rotated of a small angle and still satisfies (15). Let us now give a heuristic motivation about the technique invented in Refs. Sav03,Sav08 and developed in Ref. VSS06. Such technique is designed
85
for minimizers, that is, in the notation of (2) and Theorem 2.5, for functions u satisfying
(17) for any bounded domain
nc
and any smooth function
n (such minimizers are often named
86
transparent, since u is, locally, either constant or harmonic (further difficulties, however, are that the function has only Lipschitz junctions at the free boundary {lui = I} and that the latter is not, in principle, a smooth surface). We would like now to discuss how condition (15) may be weakened. Since u is supposed to be increasing in the nth variable and bounded, we may denote the space variable as x = (x',x n ) E ~n-l X ~ and set lim
'IT(x'):=
Xn --++00
and
lim
y'(x'):=
Xn --+-00
u(x', x n ) u(x',x n ).
(18)
Roughly speaking, 'IT and y. are the "profiles" of our solution at infinity (and they satisfy the same equation in ~n-l). As in the case of minimal surfaces, Theorem 2.3 states that the control of such profiles may give rigidity information on the solution. More generally, one can obtain the following general statement: Theorem 2.5. Let 1 < p <
+00 and
W(r) Let u : ~n
in
----+
1
-(1 - r2)p. 2p
(19)
[-1,1] be a weak solution of
with onu > Suppose that
~n,
:=
~pu - W'(u) = 0
(20)
both 'IT and y. are 2D.
(21)
o.
Then, 'IT is identically is ID.
+1
and y. is identically -1. Also, if n :::; 8, then u
We observe that the characterization of the profiles 'IT and y. in Theorem 2.5 is valid in any dimension. When n :::; 4 and p = 2 the assumptions of Theorem 2.5 may be weakened further on, since it is enough to require only that either 'IT or y. is 2D instead of (21), namely the following result holds: Theorem 2.6. Let p
= 2,
n :::; 4,
87
Let u:
]Rn ~
[-1,1] be a weak solution of
t.u - W'(u) = 0 in
with Gnu> O. Suppose that either 'IT or:g is 2D. Then, u is 1D.
]Rn,
Of course, condition (21) is weaker than (15), thus Theorem 2.5 includes Theorem 2.3. The case 1 < P < +00 under condition (15) is dealt with in Ref. VSS06, and the full statement of Theorem 2.5, as well as the one of Theorem 2.6, is obtained in Ref. FV08a. We would also like to recall that, by maximum principle-type tools, Ref. Mod85 has proved the following pointwise energy bound for global bounded solutions of t.u - W'(u) = 0, with W ;::: 0 a double well potential: (22) for any x E ]Rn. Moreover, (22) has been extended to global bounded solutions of (20) by Ref. CGS94, where it is also shown that if equality in (22) holds at a point, then the equality holds everywhere and u is 1D. Indeed, once the equality in (22) holds everywhere, the fact that u is 1D follows from the theory of isoparametric surfaces. More precisely, when equality in (22) holds everywhere, t.u and l'Vul are constant on the level sets of u: then, thanks to a full classification of such surfaces, one gets that u is ID. See Ref. DG02 for a more exhaustive discussion about this. Finally, we recall that for n = 4, 5 it is proved in Ref. GG03 that Conjecture 1.1 holds true for a special family of solutions satisfying some suitable antisymmetry conditions.
2.4. The case of the quasiminima Due to suitable density estimates and extended r -convergence techniques, FV08b the ideas of Refs. Sav03,Sav08 may be also extended to quasiminimizers. As usual, recalling the notation in (2), given Q ;::: 1, we say that u is Q-minimal if
88
for any bounded domain
nc
]Rn and any smooth function
n. Of course, the case Q = 1 reduces (23) to (17). If u:]Rn ___ [-1,1] is a weak solution of (20) in]Rn satisfying (23) and U f is as in (4), then it is shown in Ref. FV08b that U f Ltoc -converges to a step function XE - XlRn\E and that the level sets of U f approach 8E locally uniformly. Thus, the following results have been proved in Refs. FV08a, FV08b:
Theorem 2.7. Let W be as in (19) and u: ]Rn --- [-1,1] be a weak solution of (20) in]Rn satisfying (23). Suppose that one of the following conditions holds: • either: n ::; 4 and 8n u > 0, • or: 8E is a flat hyperplane and Q is close enough to 1. Then, u is ID. Analogous symmetry results hold for quasiminimal solutions whose level sets are sublinear.Fvo8a
2.5. The case in which the level sets are global graphs Level sets of monotone solutions satisfying (15) are graphs in the nth direction. Though this may not be, in principle, the general situation in Conjecture 1.1, as illustrated by (5), the question whether Conjecture 1.1 holds under an additional graph assumption for the level set is natural. Though a complete picture is not available even in this case, the following resultFV08a holds true:
Theorem 2.8. Let 1 < p < +00, W be as in (19) and u : ]Rn --- [-1,1] be a weak solution of (20), with 8n u > O. Suppose that the level set {u = c} is a complete grapM in the nth direction. Assume also that one of the following holds: • either: p = 2 and n ::; 8, • or: n ::; 4, tWe say that the level set {u = c} is a complete graph whenever there exists r lR in such a way that
{u = c} = {(x',
Xn)
E lR n -
1
x lR
S.t. Xn
= r(x')}.
: lR n -
1
-+
89
• or: p :::: n - 3, n ::; 8 and c
= o.
Then, u is ID. We remark that Theorem 2.8 comprises Theorem 2.3 as a particular case.
2.6. The fully nonlinear case
A natural question is whether any analogue of Conjecture 1.1 holds for fully nonlinear elliptic operators (see Ref. CC95a for definitions). With this respect, the following result holdspsso8 Theorem 2.9. Let F : Mat (2 x 2) ----t JR be a uniformly elliptic fully nonlinear operator, such that F(O) = o. Suppose that there exists an increasing function go E C 2 (JR, ( -1, 1)) such that
lim 90(t) = ±1
t->±oo
and
for any ~ E 8 1 and any x E JR 2 . Let u E C 2 (JR, (-1,1)) be a solution of
in JR2, with f}zu > O. Then, u is ID. The requirement on go in the statement of Theorem 2.9 reduces to the existence of a monotone ID solution. When F is the Laplacian, the proof of Theorem 2.9 given in Ref. D8808 also provides a nonvariational approach to Conjecture 1.1 when n = 2. Of course, it would be nice to investigate whether a statement analogous to Theorem 2.9 holds in higher dimension. We also recall that, for fully nonlinear equations in a suitable form, namely of the form F(div (a(lV'ul)V'u), u) = 0, the symmetry of monotone solutions in JR2 was proved in Theorem 7.4 of Ref. Far07.
90
2.7. The fractional Laplacian case Recently, the literature has payed much attention to the fractional Laplace operator. Such operator is indeed at the base of classical harmonic analysis, since it can be defined via Fourier transform
(-~)Sv(x)
:=
J- 1 (1~12S(JV)(~»).
Here, s E (0,1), J denotes the Fourier transform and ~ is the variable in the frequency space. Equivalently, the fractional Laplacian may be defined (neglecting renormalization constants) via singular integral theoryLan72,Ste70 as
r v(x) - v(y) Ix _
s
(-~) v(x) := P.V. J~n
yln+2s dy.
Here, P. V. denotes the principal value of the above singular integral (which is of course well-defined provided, say, that v E c2(1~n». Fractional elliptic operators have several important applications, since they arise as infinitesimal generators of Levy processesBer96 (also in relation with finance CT04 ), in the ultrarelativistic limit of quantum mechanics,FdlL86 in the theory of quasigeostrophic flows MT96 ,Cor98 and in water waves. CG94,NS94 A natural question in this framework is whether an analogue of Conjecture 1.1 is valid for fractional Laplacian operators. We have the following result: Theorem 2.10. Let u E C 2(]R2) be a bounded solution of (_~)su
= f(u),
with s E (0,1) and f locally Lipschitz. Suppose that [hu > O. Then, u is 1D. Theorem 2.10 has been proved in Ref. CSM05 for s = 1/2 and in Refs. SV07,CS08 for any s E (0,1), reducing the fractional Laplace operator in ]R2 to a possibly degenerate or singular boundary reaction in the halfspace]R2 x (0,+00). It would be interesting to know whether analogues of Theorem 2.10 hold in higher dimension.
2.8. The Heisenberg group case Let us consider the variables x E ]Rn, y E ]Rn, t E ]R and the vector fields Xi := OXi +2YiOt, Yi := 0Yi -2XiOt· This setting is known as the Heisenberg
91
group lHIn and the analogue of the Laplacian is the so-called Kohn Laplacian n
~lHln
=
2:)xl + Y?). i=l
The Heisenberg group is a standard noncommutative model for vector fields satisfying the so-called Hormander condition. Hor67 Thence, it is a natural question whether any analogue of Conjecture 1.1 holds in lHIn. On one hand, there are some positive results. For instance, it is known BPo2 that, if U satisfies (24) in lHIn ,
lui
S 1, and lim
U(Xl,""
Xn,
Xl---+±ex>
Yl, ... , Yn, t)
= ±1
uniformly, then U is 1D (this is an Heisenberg group analogue of Theorem 2.4 and extensions to Carnot groups have been given in Ref. BL02). Moreover, one has Bvo8 that the level sets of minimal solutions behave as codimension 1 interfaces with respect to a suitable density. Moreover, symmetry results in the Grushin plane inspired by Conjecture 1.1 have been discussed in Ref. FV08c. On the other hand, the following negative result also holds: BLo3 there exists a cylindrically symmetric solution U of (24) satisfying lui < 1, atu > 0 and lim
t-;±oo
U(Xl,""
Xn,
Yl,···, Yn, t) = ±1.
(25)
In particular, such U is not 1D and so an analogue of Conjecture 1.1 does not hold if one selects t as the monotonicity direction. It is not known if the limits in (25) are uniformly attained by the example of Ref. BL03, though a byproduct of Ref. BV08 is that the rescaling of its level sets converge locally uniformly to {t = O}. The counterexample in Ref. BL03 leaves open some questions related to Conjecture 1.1. Indeed, the direction t, which is often referred to as the "center of the Heisenberg group" , is somewhat special, since the vector field at is obtained by commuting the X and Y vector fields, hence at has the "dimension of a second derivative". With this respect, a natural vector field to look at, following Ref. BL03, is
92
Of course, Xi and Yi may be analogously defined for i = 1, ... , n and th~y are generated by infinitesimal left actions of the group. Notice that Xl commutes with both Xi and Yi, thence it commutes with ~IHIn. In this framework, an interesting analogue of Conjecture 1.1, still left open after Ref. BL03, could be to investigate whether or not bounded solutions of (24) satisfying Xl U > 0 need be 1D, at least if n is not too large. Acknowledgments We thank Luigi Ambrosio, Isabeau Birindelli, Nassif Choussoub, Changfeng Cui, Luciano Modica and Ermanno Lanconelli for their useful comments. EV was supported by MIUR Variational Methods and Nonlinear Differential Equations. References AAC01.
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95 GT99. GTOl.
Gui08. Hor67. JM04.
Lad67.
Lan67.
Lan72.
MM77. MM80.
Mod85.
Mod87.
MT96.
NS94.
Row79.
Sav03.
G. W. Gibbons and P. K. Townsend. Bogomol'nyi equation for intersecting domain walls. Phys. Rev. Lett., 83(9):1727-1730, Aug 1999. David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. Changfeng Gui. Hamiltonian identities for elliptic partial differential equations. J. Funct. Anal., 254(4):904-933, 2008. L. Hormander. Hypoelliptic second order differential equations. Acta Math., 119(1):147-171, 1967. David Jerison and Regis Monneau. Towards a counter-example to a conjecture of De Giorgi in high dimensions. Ann. Mat. Pura Appl. (4), 183(4):439-467, 2004. O. A. Ladyzenskaja. New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Trudy Mat. Inst. Steklov., 102:85-104, 1967. L. D. Landau. Collected papers of L. D. Landau. Edited and with an introduction by D. ter Haar. Second printing. Gordon and Breach Science Publishers, New York, 1967. N. S. Landkof. Foundations of modern potential theory. SpringerVerlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. Luciano Modica and Stefano Mortola. Un esempio di r- -convergenza. Boll. Un. Mat. !tal. B (5), 14(1):285-299, 1977. Luciano Modica and Stefano Mortola. Some entire solutions in the plane of nonlinear Poisson equations. Boll. Un. Mat. Ital. B (5), 17(2):614-622,1980. Luciano Modica. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math., 38(5):679-684, 1985. Luciano Modica. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal., 98(2):123142, 1987. Andrew J. Majda and Esteban G. Tabak. A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Phys. D, 98(2-4):515-522, 1996. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). P. 1. Naumkin and 1. A. Shishmarev. Nonlinear nonlocal equations in the theory of waves. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by Boris Gommerstadt. J. S. Rowlinson. Translation of J. D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density". J. Statist. Phys., 20(2):197-244, 1979. Vasile Ovidiu Savin. Phase Transitions: Regularity of Flat Level Sets. PhD thesis, University of Texas at Austin, 2003. https://dspace.lib.utexas.edu/bitstream/2152/993/1/savinov032.pdf.
96 Sav08. Sch95. Ste70.
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Ovidiu Savin. Phase transitions: Regularity of flat level sets. Ann. of Math. (to appear), 2008. M. Schatzman. On the stability of the saddle solution of Allen-Cahn's equation. Proc. Roy. Soc. Edinburgh Sect. A, 125(6):1241-1275, 1995. Elias M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J., 1970. Princeton Mathematical Series, No. 30. Berardino Sciunzi and Enrico Valdinoci. Mean curvature properties for p-Laplace phase transitions. J. Eur. Math. Soc. (JEMS),7(3):319-359, 2005. Yannick Sire and Enrico Valdinoci. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. Preprint, 2007. http://wVlW .math. utexas .edu/mp_arc. Peter Sternberg and Kevin Zumbrun. Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. Anal., 141( 4):375-400, 1998. Peter Sternberg and Kevin Zumbrun. A Poincare inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math., 503:63-85, 1998. Enrico Valdinoci. Flatness of Bernoulli jets. Math. Z., 254(2):257-298, 2006. Enrico Valdinoci, Berardino Sciunzi, and Vasile Ovidiu Savin. Flat level set regularity of p-Laplace phase transitions. Mem. Amer. Math. Soc., 182(858) :vi+ 144, 2006.
97
TWO REMARKS ON PERIODIC SOLUTIONS OF HAMILTON-JACOBI EQUATIONS HITOSHI ISHII
Department of Mathematics, Waseda University Nishi-waseda, Shinjuku, Tokyo, 169-8050 Japan E-mail: [email protected] HIROYOSHI MITAKE
Department of Pure and Applied Mathematics Waseda University Ohkubo, Shinjuku, Tokyo, 168-8555 Japan E-mail: [email protected] We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation Ut + H(x, Du) = f(t) in 0 x JR, where 0 is a bounded domain of JRn, with the Dirichlet boundary condition u = 9(X, t) and that of a subsolution of the stationary problem H(x, Dv) = (f) under the assumptions that the function (f(t), g(x, is periodic in t and H is coercive. Here (f) denotes the average of f over the period. This proposition is a variant of a recent result for 0 = JRn due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, Ut + H(x, Du) = 0 in 0 x JR and u = 9 on ao x JR, is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.
t»
Keywords: Hamilton-Jacobi equations; periodic solutions; Aubry sets.
1. Introduction
In this paper we consider the Hamilton-Jacobi equation with the Dirichlet boundary condition: Ut(x, t)
+ H(x, u(x, t), Du(x, t)) = f(t)
{ u(x, t) = g(x, t)
for (x, t) E
for (x, t) E
n x JR,
an x lR.
n
(D)
Here n is a bounded domain of JRn, u = u(x, t) is a function on x JR which represents the unknown function, H is a given function on x JR x JRn which
n
98 is the so-called Hamiltonian, f is a given periodic function with period T > 0, 9 = g(x, t) is a given function on an x JR, the functions u, H, f and 9 are scalar functions and Ut and Du denote the derivatives au/at and (aU/ aX 1, ... ,au/axn ), respectively. In this note we will be concerned only with viscosity solutions of Hamilton-Jacobi equations and thus we call them just solutions. The boundary condition, u = 9 on an x JR, is also understood in the viscosity sense. We refer to [1,2,6J for overviews on viscosity solutions theory. We assume throughout the following (Al)-(A5): (AI) (A2) (A3)
HE BUC(O x [-R, RJ x B(O,R)) for all R H is monotone. That is, for each (x,p) E H(x,·, p) is non-decreasing on R H is coercive. That is, for every r E JR,
lim H(x, r, p) ipi->oo (A4) (A5)
=
00
> 0. 0 x JRn,
the function
uniformly for x E O.
0. is a bounded, open, connected subset of JRn with CO boundary. 9 E BUC(an x JR).
Here and in what follows BUC(X) denotes the space of bounded, uniformly continuous functions on metric space X. In (A4), it is assumed that 0. has CO boundary. We mean by this CO regularity that for each z E an there are a neighborhood V of z and a C 1 diffeomorphism of V to B(O, r), with r > 0, such that (z) = and (n n V) = B(O,r) n {(x',x n ) E JRn-1 x JR : Xn > h(x')} for some h E C(JRn-1). The role of (A4) in this note is to guarantee (see [19]) that if u E C(n) satisfies IDu(x)1 ~ C in 0. in the viscosity sense for some constant C > 0, then u is uniformly continuous on 0., so that it can be extended uniquely to 0 as a continuous function. In this note we often deal with soloutions u(x, t) of (D) which is periodic in t. Given a set X and a function w = w(x, t) defined in X x JR, we call w periodic with period T if it is periodic in t with period T for all x EX. In the next section we assume that the function g(x, t) is also periodic with period T and establish the equivalence between existence of a periodic solution u E C(O x JR) of (D) with period T and that of a subsolution of
°
H(x, v, Dv) = (J)
in
n.
(8)
The result here gives a variant of [5,Theorem 4.1], due to Bostan-Namah, where 0. is replaced by JR n . Our proof is somewhat simpler than the one in [5]. Our result covers also the equivalence between existence of a bounded
99 solution of (D) and that of a subsolution of (S) without the assumption that 9 is periodic. We refer for instance to [23, 22, 17] for some existence results of periodic solutions of Hamilton-Jacobi equations. In the final section, Section 3, we restrict ourselves to the case where f = 0, H = H(x,p) and H(x,p) is convex in p. We establish a theorem on representation of bounded solutions u of (D) and then show under some additional assumptions that if 9 is periodic with period T, then any solution u(x, t) of (D) is constant in t on the Aubry set A. Actually, we show this constancy result under an assumption more general than the periodicity of g. We give the definition of the Aubry set A in Section 3, following [11-13,19,20]. The constancy in t of periodic solutions u(x, t) of (D) on the Aubry set indicates a new characteristic of the Aubry set for problem (D) in the periodic setting in t, and indeed has an important role in the dynamical approach to the asymptotic behavior for large t of solutions of the CauchyDirichlet problem for Hamilton-Jacobi equation Ut +H(x, Du) = 0 with the Dirichlet condition u = g. See [22] for the asymptotic behavior of solutions of the Cauchy-Dirichlet problem with periodic Dirichlet data in t. We refer to [8-13] for Aubry sets and weak KAM theory.
2. Existence of periodic solutions Throughout this section we assume that f E C(JR) is a periodic function with period T > O. The following theorem is our main result in this section. Theorem 2.1. (i) Problem (D) has a bounded solution u E C(O x JR) if and only if (S) has a subsolution v E C(O). (ii) Assume that 9 is periodic with period T. Then (D) has a periodic solution u E C(O x JR) with period T if and only if (S) has a subsolution v E C(D). We set
1 t
F(t) =
(l(S) - (I))ds
for t E lR.
Note that F is a C 1 periodic function on JR with period T. Also, we set G(x, t)= g(x, t) - F(t) for (x, t) E an x lR. We consider the problem
Wt(x, t)
+ H(x,
+ F(t), on an x lR.
w(x, t)
{ w(x, t) = G(x, t)
Dw(x, t)) = (I)
in
nx
JR,
(D')
100
Lemma 2.1. (i) Problem (D) has a bounded solution u E C(n x JR.) if and only if (D') has a bounded solution w E C(n x JR.). (ii) Problem (D) has a periodic solution u E C(n x JR.) with period T if and only if (D') has a periodic solution w E C(n x JR.) with period T. Proof. Observe that if u is a solution of (D), then w(x, t) := u(x, t) - F(t) is a solution of (D'). On the other hand, if w is a solution of (D'), then u(x, t) := w(x, t) + F(t) is a solution of (D). Note that u is bounded on x JR. if and only if so is the function w(x, t) := u(x, t) - F(t). Note also that u is periodic with period T if and only if so is the function w. 0
n
It is a useful and classical observation on solutions of (D) or (D'), which is a consequence of the coerciveness of the Hamiltonian H, that if u is an upper semi-continuous subsolution of (D) or (D') on x (a, b), then u(x,t)::; g(x,t) on an x (a, b).
n
Proof of Theorem 2.1. First of all, assume that (D) has a bounded continuous solution. By Lemma 2.1, there is a bounded solution wE C(n x JR.) of (D'). Let M > 0 be a constant such that [w(x, t)[ V [F(t)[ ::; M for all (x, t) E x R 8et
n
v(x) = supw(x, t)
for x En,
tEIR
and observe by the stability of viscosity property that u := v* is a subsolution of H(x, u - M, Du)
= (J)
in
n,
where v* denotes the upper semi-continuous envelope of v. 8ince H is coercive and n has CO boundary, we find that v* E C(n) and v* - M is a subsolution of (8). Next, we suppose that (8) has a subsolution v E C(n) and show in view of Lemma 2.1 that (D') has a bounded solution w E C(n x JR.). In view of the monotonicity of H, we see that the function v + C, with any negative constant, is a subsolution of (8). By adding a negative constant to v if necessary, we may assume that v(x) ::; G(x, t) for all (x, t) E an x R It is obvious that v - M is a subsolution of (D'). In view of the coerciveness of H, we may choose a supersolution 'lj; E C(n) of (8) so that 'lj;(x) 2: G(x, t) for all (x, t) E an x R Note that 'lj; + M is a supersolution of (D'). We define w : x JR. ----) JR. by
n
w(x, t) = sup{ ¢(x, t) : ¢ is a subsolution of (D'), ¢(x, t) ::; 'lj;(x)
+M
for all (x, t)
En x JR.}.
101
It is clear that v(x) ~ w(x, t) ~ ,¢(x) + M for all (x, t) E n x R In view of the Perron method we see that the upper semi-continuous envelope w* (respectively., the lower semi-continuous envelope w* ) of w is a subsolution (respectively., supersolution) of (D'). In particular, since w*(x, t) ~ ,¢(x) + M for all (x, t) En x JR., we find that w* ~ wand hence w* = w on n x R Also, it is clear by the definition of w that if 9 is periodic with period T, then the function w is periodic with period T. It remains only to show that w E C(n x JR.). Let w be a modulus such that
IG(x,t) - G(x, 8)1 V IF(t) - F(8)1
~
w{lt - 81)
for all t,8 E JR., x E an.
For any h E JR. we consider the function wh(x, t) := w(x, t + h) - w(lhl). Observe that w h is a subsolution of (D') and satisfies wh(x, t) ~ ,¢(x) + M for all (x, t) En x R Therefore, by the definition of w, we have wh(x, t) ~ w(x, t) for all (x, t) En x R That is, we have w(x, t + h) ~ w(x, t) + w(lhl) for all (x, t, h) En x JR. x R Hence we get
Iw(x, t) - w(x, 8)1 ~ w(lt - 81) for all t, 8 E JR., x E n.
(1)
We suppose for the moment that IG(x, t) - G(x, 8)1 ~ Lit - 81 for all (x, t, 8) E an x JR.2 and for some L > o. Then, since F E C1(JR.), we may assume by replacing L by a larger number if necessary that
IG(x,t) - G(x,8)1 V IF(t) - F(8)1
~
Lit - 81
for all t,8 E JR., x E an.
This combined with (1) ensures that Iw(x, t) - w(x, 8)1 x E nand t, 8 E JR., from which we infer that
H(x, w(x, t) - M, Dw(x, t))
~
L
+ (I)
in
~
Lit - 81 for all
nx R
The coerciveness of H and the CO boundary regularity of n guarantee that -2 Iw(x, t) - w(y, t)1 ~ wo(lx - yl) for all (x, y, t) E n x JR. and for some modulus w00 Thus, w is uniformly continuous on n x R Now, we treat the general situation where 9 E BUC(an x JR.). We approximate G by a sequence of functions Gk(X, t), k = 1,2, ... , such that G(x, t) - 11k ~ Gdx, t) ~ G(x, t) and IGk(x, t) - Gk(x, 8)1 ~ Lkl t - 81 for all (x, t, 8) E an x JR. x JR. and for some Lk > o. Consider the problem Zt {
+ H(x, z + F,
z(x, t) = Gk(X, t)
Dz) = (I)
in n x JR.
for (x, t) E an x R
(D",)
102
We define Wk : IT x JR
--+
JR by
Wk(X, t) = sup{ 4>(x, t) : 4> is a subsolution of (Dk), 4>(x, t) :::; '¢(x) + M for all (x, t)
E
IT x JR}.
We have already observed that Wk E C(IT x JR) since the family of functions Gk(x, .), with x E IT, are equi-Lipschitz continuous on JR and that Wk is a solution of (Dk). Since G k :::; G, by the definition of w, we see that Wk :::; W in IT x R Similarly, we see that W - 11k:::; Wk in IT x R Thus we see that W is a uniform limit of a sequence of functions in BUC(IT x JR). Hence, we 0 have wE BUC(IT x JR). Remark 2.1. (i) The above proof shows that if there is a subsolution of (S), then there is a solution U E BUC(IT x JR) of (D). Moreover the solution constructed in the above proof is the maximal solution of (D) in the sense that it is the pointwise maximum of all subsolutions of (D). (ii) The periodicity of J can be replaced by its almost periodicity in Theorem 2.1. In the case of almost periodic J, we have to modify the definition of (f) and to replace it by
liT
(f) = lim T->oo
T
J(t)dt.
0
3. Constancy on Aubry sets
In this section we always assume that H(x, r,p) does not depend on rand J = o. We write H(x,p) for H(x,r,p). Our problems (D) and (S) thus read Ut {
u
+ H (x, Du) = =9
on
0
in
n x JR,
an x JR
(D)
and
H(x, Dv)
=0
in
n.
(S)
We investigate here properties of bounded solutions of (D). Thus, in view of Theorem 2.1 (i), we make the following assumption:
(A6)
There is a subsolution of (S).
In addition to (A1)-(A6), we assume the following throughout this section:
103
(A7)
H is convex. That is, the function p x E IT.
f-->
H(x,p) is convex for any
(AS) Either of the following (AS)+ or (AS)_ holds: (AS)+ There exists a modulus w satisfying w(r) > for all r > Osuch that for all (x,p) E IT x lRn such that H(x,p) = and for all
°
~
E D:; H(x,p), q E lR n ,
H(x,p + q)
~
°
H(x,p) + ~. q + w((~· q)+)
where D:; H(x,p) stands for the sub differential of H with respect to the second variable p and r + := max{O, r} for r E lR. (AS)_ There exists a modulus w satisfying w(r) > for all r > Osuch that for all (x,p) E IT x lRn such that H(x,p) = and for all
°
~
ED:; H(x,p), q E lR n , H(x,p + q)
~
°
H(x,p) + ~. q + w((~· q)_)
where r _ := max{O, -r} for r
lR.
E
We remark that condition (AS) appears in the study of the asymptotic behavior of solutions of the Cauchy problem for Hamilton-Jacobi equations. For this, see [14-16] and also [3]. Condition (AS) is a sort of strict convexity requirement on H at the level of H = 0. Let L denote the Lagrangian of H, that is,
L(x,O= sup(~·p-H(x,p))
for(x,~)EITxlRn.
pEIRn
Define the functions e : IT x IT x (0,00) b : IT x lR ----+ lR by
e(x,y,t) = inf
{lot Lb]
----+
lR U {oo}, d : IT x IT
----+
lR and
: "( E AC([O, t],IT) , "((t) =x, "((0) = y},
d(x, y) = inf e(x, y, t), t>O
b(x,t) = inf {e(x,y,T)
+ g(y,t - T) : T > 0,
y E aD}.
Here and henceforth AC([a, b], IT) denotes the space of absolutely continuous functions on [a, b] with values in IT. Also, we use the abbreviated notation Lb] to denote the integral L("((s), -y(s))ds. We extend the domain of
f:
f:
definition of e to IT2 x [0,(0) by setting e(x,x,O) = x -=I- y. Recall that
°
and e(x,y,O) = 00 if
d(x,y) = sup{v(x) - v(y) : v E C(IT) is a subsolution of (S)}.
104
See [7,12,13J for similar results on the n-dimensional torus. We note by The validity (A6), the coerciveness (A3) and this formula that d E of this formula for d can be seen as follows. Let w(x, y) denote the right hand side of the above formula. It follows from [20, Proposition 5.1J that v(x) v(y) ::; d(x, y) for any x, yEn if v is a subsolution of (S). Hence, we have w(x, y) ::; d(x, y). On the other hand, since sUPnxB(O,
c(n\
H(x, Du(x)) ::; 0 in { H(x, Du(x)) 2: 0 on
s:
(SC)
n.
We now introduce the (projected) Aubry set A for H, associated with (SC), by setting
A = {y En: d(·, y) is a solution of (SC)}. We refer to [19,20J for related observations and to [11-13J for general properties of Aubry sets. In particular, it is known (see [20, Proposition 6.4]) that A :f: 0 if and only if (SC) has a solution and that A is a compact set. Also it is known that, under hypothesis (A6), A :f: 0 if and only if (SC) has a supersolution. Also, the following characterization is a classical and crucial observation regarding Aubry sets. Let T > O. A point yEn is in A if and only if inf {
lot Lb] : t 2:
T, "(
E AC([O,
t],O'), "((0)
= "((t) =
y}
= O.
(2)
See [18, Proposition A.3] or [12, Theorem 4.3] for a proof of this characterization. The function e is a "fundamental solution" of the state-constraint problem for Ut + H(x,Du) = 0 in 0' x (0,00). Indeed, for any f E C(D), the
105
solution u of the Cauchy problem of
°
Ut + H(x,Du) ~ in n x (0,00), Ut+H(x,Du) ~o in nx (0,00), { u(·,O) = f, can be written (see [20, Eq. (5.1)]) as
u(x, t) = inf{f(y)
+ e(x, y, t)
: YEn}. 2
Lemma 3.1. (i) The function e is bounded below on n x [0, (0). (ii) e 2 is a lower semi-continuous function on n x [0,(0). (iii) For each yEn the function u := e(·,y,·) is a solution of the state-constraint problem for Ut + H(x, Du) = in n x (0, (0) in the sense of Barron-Jensen [4]. That is, for any (x,t) En x (0,00) and ¢ E C 1 (n x (0,00)), ifu - ¢ attains a minimum at (x, t), then
°
¢t(x, t) { ¢t(x, t)
+ H(x, D¢(x, t)) = + H(x, D¢(x, t)) ~
° °
if x E if x E
n, an.
Remark 3.1. In the following presentation, we will not use the above assertion (iii). 2
Proof. In this proof we set Q = n x (0, (0). By the definition of d, we have d(x, y) ~ e(x, y, t) for all (x, y, t) E Q, with t > 0. Clearly, we have d(x,y) ~ e(x,y,O) for all X,y En. Thus, e is bounded below on Q. To see that e is lower semi-continuous on Q, we fix any (x, y, t) E Q and assume that there is a sequence {(Xk' Yk, tk)hEN C Q such that limk-+oo(Xk, Yk, tk) = (x, y, t) and limk-+oo e(xk, Yk, tk) = eo for some eo E R We choose a sequence of curves "Ik E AC([O, tk], n) such that for all kEN, "Ik(tk) = Xk, "Ik(O) = Yk and
e(xk' Yk,tk)
+ ~ > ltk L["Ik].
By using [18, Lemmas 6.3, 6.4], we deduce that there is a curve "I E AC([O, t], n) such that "I(t) = x, "1(0) = Y and J~ L["I] :::: eo· Hence, we get e(x, y, t) ~ eo, which shows the lower semi-continuity of e at (x, y, t). We remark here that the variational problem t
e(x, y, t) = inf{l Lb] : "I E AC([O, t], n), "I(t) = x, "1(0) = y} has a minimizer for every (x, y, t) E Q.
106 2
Next, we show that e is lower semi-continuous at points in 0 x {OJ. Fix any R > 0 and a constant C R > 0 so that H(x,p) ::; C R for all (x,p) E OX B(O, R). Also, fix any yEO. The function w(x, t) = Rlx - yl - CRt of (x, t) on 0 x [0,00) is a subsolution of Wt + H(x, Dw) = 0 in n x (0,00). Due to [20, Proposition 5.1], we obtain
w({(t), t) ::; w({(O),O)
+ lot L[r]
for any t > 0 and, E Ae([O, t], 0). From this we get
e(x, y, t) ;::: Rlx - yl - CRt
for all (x, y, t) E Q.
-2
Thus, for any (xo, Yo) En, we obtain lim inf
e(x, y, t) ;::: Rlxo - Yol.
(x,Y)->(XO,YO), t->O+
As R > 0 is arbitrary, we see that lim inf (X,Y)->(xo,YO), t->O+
e(x, y, t) ;::: e(xo, Yo, 0).
This completes the proof of the lower semi-continuity of e on Q. Now, we fix yEO and set u(x, t) := e(x, y, t) for (x, t) E 0 x (0,00). Let ¢ E C 1 (0 x (0,00)) and assume that u - ¢ attains a strict minimum at (x,l). We choose a minimizer, E AC([O,~,O) for e(x,y,l), i.e., the curve , has the properties: ,(l) = x, ,(0) = y and
e(x, y, l) = lof L[r]. We need to show that
¢t(:i, l) + H(x, D¢(x, l)) ;::: 0
(3)
¢t(x,l)+H(x,D¢(x,l))::;O ifxEn.
(4)
and also
We suppose the contrary of (3), i.e, ¢t(X, l) + H(x, D¢(x, l)) < o. We choose an E E (0, l) so that ¢t(x, t) + H(x, D¢(x, t)) ::; 0 in B(x, E) x [f - E,~. We select 7 E [f - E, l) so that ,(8) E B(X,E) nO for all 8 E [7, ~ and either 7 = f - E or ,(7) E OB(X,E) \ O. We may assume by adding a constant to
107
¢ that u(x, l) = ¢(x, l). Then we have u > ¢ on (8B(x, c) nIT) x [f B(x, c) x {f - c}. We observe that ¢(r(l) , l) - ¢(r(T), T) =
: ; it
(¢t(r(s),s)
it
(¢t(r(s), s)
+ D¢(r(s), s) . 1'(s))
+ H(r(s), D¢(r(s), s)) + L(r(s),1'(S)))
and therefore
u(x,l) = ¢(x,l) < U(r(T),T)
+
it
Lb] =
10,
~
u
ds
ds:::;
it
Lb]
lot Lb] = e(x,y,l) = u(x,l).
This is a contradiction, which shows that (3) is valid. We next prove inequality (4). We assume that x E n. We suppose, contrary to (4), that ¢t(x, l) + H(x, D¢(x, l)) > o. We choose an r > 0 so that B(x, r) c nand ¢t(x, t) + H(x, D¢(x, t)) 2 0 in B(x, r) x [f, f + r]. As before, we assume that u = ¢ at (x, l). Note that u(x, t) > ¢(x, t) if (x,t) =1= (x,l). As in the proof of [18, Theorem A.l]' we find a curve 1] E AC([f, T], n), with f < T :::; f + r, such that for a.e. s E (f, T),
D¢(1](s), s) . ij(s) = L(1](s), ij(s)) By replacing
1](s)
E
T
+ H(1](s), D¢(1](s), s)).
by a smaller number (> l) if necessary, we may assume that E [f,T]. We now compute that
B(x,r) for all s
U(1](T),T) > ¢(1](T),T)
iT + iT
= ¢(x, l) + =
u(x,l)
(¢t(1](S), s) L[1]] =
+ H(1](s), D¢(1](s), s)) + L(1](S), ij(s)))
ds
lot Lb] + iT L[1]] 2 U(1](T),T).
This is a contradiction, from which we conclude that (4) is valid. Lemma 3.2. (i) There is a constant Co > 0 and for each (z, T) E a neighborhood V of z, ralative to IT, such that
e(x,y,T + t) :::; CO(T
+ t) for all X,y
E
V, t
0
ITx(O, 00)
2 o.
(ii) There are constants T1 > 0 and C 1 > 0 such that e(x, y, Tr) :::; C 1 for -2 all (x,y) En. Proof. As noted before, there are constants J > 0 and C > 0 such that L(x,~) :::; C
for all (x,~) E IT x B(O, J).
(5)
108
For any (x,t) E IT x (0,00), if we set ,(8):= x, then
e(x, x, t) ::;
1t
L[,] ::; Ct.
(6)
We note that for any x, y, ZEIT and t, 8 ~ 0,
e(x, y, t + 8) ::; e(x, z, t) + e(z, y, 8).
(7)
We show that assertion (i), with Co = C, holds. To see this, we fix r > 0 and ZEIT. In view of (6) and (7), we need only to prove that there is a constant p > 0 such that
e(x, y, r) ::; Cr
for all x, y
E B(z, p) nIT.
(8)
According to [20, Lemma 4.2]' there exists ( E coo(IT,lRn) such that En for all (x,c:) E IT x (0, 1]. Choose a constant M > 0 so that max?! 1(1 ::; M and Jr::; 3M. Set t = r/3 and 'T}(x) = (J/M)((x) for x E IT. Note that (Jt/M) ::; 1 and max?! I'T}I ::; J. In particular, we have x + 8'T}(X) E n for all (X,8) E IT x (O,t]. Select r > 0 so that B(z + t'T}(z),r) en and also 2r ::; M. Next, in view of the continuity of 'T}, we choose p > 0 so that x + t'T}(x) E B(z + t'T}(z) , r) for all x E B(z, p) nIT. Fix any x, y E B(z, p) n IT. We define the curve, E AC([O, r], IT) by concatenating three line segments [y, q], [q, p] and [p, x], where p := x + t'T}(x) and q := y + t'T}(y) , as follows:
x
+ c:((x)
,(8) =
y+8'T}(Y) q + (s~t) (p - q)
forO::;8
{ x + (3t - 8)'T}(X) for 2t ::; 8 ::; 3t. lt is clear that, E AC([O, r],IT), ,(0) = y and ,(r) = x. Noting that Ipql/t::; 2r/t::; J, we see that h(8)1 ::; J for a.e. 8 E (0, r) and consequently
e(x, y, T) ::;
1T Lb] ::; CT,
which completes the proof of (i). Next we show that assertion (ii) is valid. Let Co > 0 be the constant from assertion (i). According to assertion(i), for each zEIT, we may choose an open neighborhood Vz of z such that
e(x, y, 1) ::; C o(l
+ t)
for all x, y
E Vz nIT,
t ~ O.
(9)
By the compactness of IT, we may choose a finite points Zl, ... , Zk E IT such that {VZj }j=l covers IT. We fix any x, y E IT. We use the connectedness of IT, to see that there is a sub-family {WI, ... , Wm } of {VZj }, with m ::; k, such
109
that x E WI, Y E Wm and Wi n Wj+l nn i= 0 for all j = 1, ... ,m -1. We select a sequence {xi }j=1 1 so that xi E WinWj+lnn for all j = 1, ... , m-1. We observe by (9) and (7) that for any t ~ 0, e(x, y, m
+ t)
::; e(x, Xl, 1)
::; Co(m
+ e(xl' X2, 1) + ... + e(xm-l, y, 1 + t)
+ t).
Therefore, in general, we have -2
for all (x,y) En.
e(x,y,k) ::; Cok
Thus, assertion (ii) is valid with T1 = k and C 1 = Cok.
o
Lemma 3.3. The function b is bounded, uniformly continuous on n x JR and it is a solution of (D). Moreover we have for all (x, t) En x JR, b(x, t)
= max{ vex, t)
: v is a subsolution of (D)}.
(10)
Proof. By Lemma 3.1 (i), the function e is bounded below. Therefore, we see that b is bounded below on n x R Indeed, we have b(x, t) ~ info2 d + infanxlR 9 for all (x, t) En x R Next, by Lemma 3.2 (ii), there are T1 > 0 and C 1 > 0 such that e(x, y, Tl) ::; C 1 for all x, yEn. Note that b(x, t) ::; inf{e(x, y, Tl)
::; C 1
+
sup 9 anxIR
+ g(y, t
- TI) : yEan}
for all (x, t) En x R
Thus, the function b is bounded on n x R Next, we show that b is the maximal solution of (D), i.e, we show that (10) holds. We write u(x, t) for the right hand side of (10). According to the proof of Theorem 2.1 (i) and Remark 2.1 (i), u is a solution of (D) and u E BUC(n). We regularize u by sup-convolutions in t as follows. Let e > 0 and set ue:(x,t) = sup (u(X,s) -
It - S12) 2e
sEIR
for (x,t) En x
R
As is well-known, the function u€ is a subsolution of Ut + H(x, Du) = 0 in n x JR and has the distributional first derivatives in L=(n x JR). Moreover lu€(x, t) - u(x, t)1 ::; wee) for all (x, t) En x JR, where w is a modulus. In particular, we have u€(x, t) ::; g(x, t) + wee) for all (x, t) E an x R Let (x,t) En x JR and T > 0, and fix any, E AC([O,T],n) such that ,(T) = x and ,(0) E an. We apply [20, Proposition 5.1], to get u€(x,t) ::;U€h(O),t-T)
+
1T Lb] ::;gh(O),t-T)+W(e) + 1T LbJ,
110
from which we deduce that u€(x, t) :::; b(x, t) + w(c). Moreover, since € > 0 is arbitrary, we find that u :::; b on x R Now, let b* denote the upper semi-continuous envelope of b. If b* is a subsolution of (D), then we have b* :::; u on x lR by the definition of u, which implies that u = b. Thus we only need to show that b* is a subsolution of (D). But, it is a classical observation (see [18, Theorem A.l]) that b* is a subsolution of Vt + H(x, Dv) = 0 in n x R Hence, it is enough to show that b* :::; 9 on an x R We fix any (y,s) E an x lR and c > O. Let Co > 0 be the constant from Lemma 3.2 (i). We choose 0 > 0 so that maxr E[s-28, s1 g(y, r) < g(y, s) + c and Coo < c. By Lemma 3.2 (i), there is a neighborhood V of y, relative to such that
n
n
n,
e(x,y,t):::; Cot
E
V x [0,(0).
+ 0], we obtain :::; e(x, y, 0) + g(y, t - 0) < Coo + g(y, s) + c :::; g(y, s) + 2c,
For any (x, t)
b(x, t)
for all (x,t)
E V
x [s - 0, s
which ensures that b*(y, s) :::; g(y, s)" Hence, b* :::; 9 on an x lR, and b* is a 0 subsolution of (D). The following two theorems are the main results in this section.
Theorem 3.1. Let u be a bounded solution of (D) on n x R Set
u_(x) = liminfu(x,t) t----?-oo
uo(x) = inf{ u_ (y)
for x
+ d(x, y):
E
n,
yEA}
for x E
n.
Then u(x, t) = uo(x) 1\ b(x, t)
for all (x, t)
En x R
(11)
n
In the above theorem, if A = 0, then uo(x) = 00 for all x E and hence (11) asserts that u = bon x R It is standard observations that uo(x) = u_(x) for all x E A and u_(x) :::; uo(x) for all x and that u_ is a solution of H(x, Du(x» = 0 in nand Uo is a solution of (SC). Here the convexity of H is essential to conclude that u_ is a subsolution of H(x, Du) = 0 in n.
n
En
Theorem 3.2. Let u and Uo be as in Theorem 3.1. Assume that liminf g(x, t) = inf g(x, t) t--->-(X)
tEIR
for all x
E
an.
Then u(x, t) = uo(x)
for all (x, t)
E
A x R
(12)
111
A consequence of the above theorem is that if g(x, t) is almost periodic in t for every x E IT, then condition (12) is satisfied and hence u(x, t) is constant in t for any x E A. In particular, if 9 is periodic, then any bounded solution u(x, t) of (D) is constant in t on the Aubry set A. A general observation on (D) is that the value of any solution u E C(IT x JR) of (D) at (x, t) E IT x JR is represented as u(x,t) = inf{e(x,y,t-s) +u(y,s) : yEIT} !\inf{e(x,y,t - 7)
+ g(y,7)
: y E &0" s < 7 < t},
(13)
where s E (-00, t) is an arbitrarily fixed number. For a proof of this formula we refer to [21, Theorems 4.1,4.3]. Let v E C(IT) be a solution of (SC). A curve I E C(( -00,0], IT) is said to be extremal for v if, for any -00 < s < t :=; 0, I is absolutely continuous on [s, t] and satisfies
it
Lb] = v(r(t» - v(r(s».
Let o:(r) denote the alpha-limit set of a curve I E C( (-00, 0], IT). That is,
o:(r)
:=
=
{y
E
IT : there exists a sequence tj
n,(C
---+
00 such that ,( -t j
) ---+
y}
-00, t]).
tEIR
It is easily checked by recalling (2) that if I is an extremal curve for some solution of (SC), then o:(r) C A. Lemma 3.4. Assume that A -I- 0. Let Uo be the function from Theorem 3.1. Let I be an extremal curve for Uo. Let E > o. Then there are a constant 70 > 0 and a neighborhood W of o:(r), relative to IT, and for each x, yEW a curve ry E AC([-7, 0]' IT), with 0 < 7 :=; 70, such that ry(O) = x, ry( -7) = Y and
Proof. By Lemma 3.2 (i) and the compactness of IT, we may choose constants r > 0, 7 > 0 and, for each ZEIT and x, y E B(z, r) n IT, a curve ~ E AC([ -7, 0], IT) such that ~(O) = x, ~(-7) = y and
1: L[~]
<E.
112
Here, since Uo E C(O), we may assume by replacing r > 0 by a smaller positive number if necessary that [uo(zd - UO(Z2)[ S; c if Zl, Z2 E 0 and [Zl - Z2[ S; 2r. Accordingly, we have
[OT L[e] < 2c + uo(x) -
uo(y). -2
Now, we set K = o{y) x o{y). Note that K is a compact subset of 0 . Let (p, q) E K and consider the neighborhood V := (B(p, r)nO) x (B(q, r)n -2 0) C 0 of (p, q). Fix any x, y E V. Since p, q E o{y), we may choose numbers 0 < tp < tq < 00 so that ,( -tp) E B(p, r) and ,( -tq) E B(q, r). By the previous observation, there are curves 6 E AC([-7, 0], 0) and 6 E AC([-7, 0], 0) such that 6(0) = x, 6(-7) = ,(-tp), 6(0) = ,(-tq) and 6( -7) = Y and such that
[OT L[6] < 2c + uo(x) -
uo( ,( -tp»,
and
[OT L[6] < 2c + uo(r( -tq » -
uo(y).
Next, we concatenate three curves 6, , and 6, to define the curve TJ. That is, we define the curve TJ E AC([-tpq, 0],0), with tpq = 27 + tq - tp, by setting for - 7 < s S; 0, for - 7 - tq + tp < for - tpq S; s S; - 7
S
S;
-7,
-
tq
+ tp.
The curve TJ has the properties: TJ(O) = x, TJ( -t pq ) = y and
[~
L[TJ] "
=
[OT L[6] + [~tp Llr] + [ : L[6] q
< 4c + uo(x) - uo(y).
For each (p, q) E K we fix tp and tq as above. Due to the compactness of K, we may find a finite sequence {(Pi,qi)}~l C K such that the family {B(Pi, r/2) x B(qi, r/2)}~1 covers K. We choose a constant 8 > 0 so that m
(a(r)+B(0,8») x (a(r)+B(0,8»)
c UB(Pi, r) x B(qi, r), i=l
and set W = (a(r) + B(O, 8») nO. Clearly, W is a neighborhood of a(r) relative to O. Also we set 70 = 27 + max1::;i::;m(tqi - tpJ. It now follows
113
that for each (x,y) E W x W there is a curve", E AC([-t, o],n), with x, "'( -t) = y and
o < t :S 70, such that ",(0) =
1:
L[",] < 4c + uo(x) - uo(y),
which was to be proven.
0
Lemma 3.5. Let 7 E lR and set I = (-00, r). Let u, v E BUC(n x 1) be a subsolution and a supersolution of (D) in x I, respectively. Assume that u :S v on A xI. Then u :S v on xI.
n
n
Remark 3.2. It follows from the above lemma that if A = 0 and if u, v E BUC(n x 1) are a subsolution and a supersolution of (D) in n x I, respectively, then u :S v on n x R In particular, if A = 0, then problem (D) has a unqiue solution in BUC(n x lR). For existence of such a solution of (D), see Remark 2.1. Proof. Fix any c E (0, 1), and set u€(x,t) = u(x,t) - c for (x,t) En x I. There is a compact neighborhood K€ of A, relative to such that u€ :S v on K€ x I. (Needless to say, we take K€ = 0 if A = 0.) As a basic property of the Aubry set, there is a function 1jJ E C(n) and, for each compact neighborhood K of A, a constant 15 K > 0 such that H(x, D1jJ(x)) ::; -15 K in n\K in the viscosity sense. For this property, see the proofs of [20, Theorem 3.3] and [12, Proposition 6.1]. We write 8£ for 15 K ,. We may assume, by adding a constant to 1jJ if necessary, that 1jJ(x) + 1 :S infnxI u for all x E so that 1jJ(x) ::; u£(x, t) for all (x, t) E x I. Accordingly, the function w(x, t) := 1jJ(x) is a solution of
n,
n,
n
+ H(x, Dw(x, t)) :S -8£ in (0, \ K£) x I, w(x, t) :S g(x, t) on (an \ K£) x I.
Wt {
We may assume, by translation if necessary, that r :S O. Fix any A E (0, 1) and choose a constant Vo > 0 so that AVO :S (1 - A)8£. For any EV and z on x I by setting V E (0, vo) we define the functions u
n
u EV (x, t) = u£ (x, t)
+ vt, z(x, t) = AUEV(X, t) + (1
- A)1jJ(X).
It is easily seen that u EV and z are, respectively, a solution of
urv + H(x, DuEV(x, t)) :S V in (0, \ K£) x I, { u£V(x, t) :S g(x, t) on (an \ K£) x I
114
and a solution of
+ H(x, Dz(x, t» ::::; )..v - (1 - )")8,, z(x, t) ::::; g(x, t) on (an \ K,,) x I.
in (n \ K,,) x I,
Zt {
Note here that uniformly for x
(1 - )")8,, ::::; 0 and also that limt->-oo z(x, t) = O. Also, observe that
)..V -
E
z(x, t) ::::; max{u"V(x, t), '¢(x)} ::::; u"(x, t)
-00
for all (x, t) EO x I.
Now, we choose a constant tv < T so that z(x, t) ::::; v(x, t) for all 0 x (-00, tv]. We apply a comparison theorem to z and v on the set (0\ K,,) x [tv, T), to find that z ::::; v on (0 \ K,,) X [tv, T). Hence we have z ::::; v on x I. That is, we have
o
)..(u(x, t) Sending v
-+
E:
+ vt) + (1
0 and then
- )..)'¢(x) ::::; v(x, t)
E: -+
0, )..
Lemma 3.6. Assume that A =I-
-+
for all (x, t) EO x I.
I, we find that u ::::; v on
0
x I.
0. Let u and uo be as in Theorem
D 3.l.
Then u(x, t) ::::; uo(x)
for all (x, t) EO x R
Proof. Fix any (x, t) E 0 x R There is an extremal curve, for uo such that ,(0)= x. See [20, Theorem 6.1] for existence of such a curve. Fix any E: > O. Let TO > 0 and W be those from Lemma 3.4. Fix a point y E 0:(/) and a sequence tj -+ 00 such that limj->oou(y,t - tj) = u_(y). Note here that yEA and hence u_ (y) = Uo (y). By passing to a subsequence if necessary, we may assume that ,( -tj) E W, ,( -tj+TO) E Wand tj > 2To for all j E N. We now assume that (AS)+ is satisfied. According to Lemma 3.4, for each j E N there is a curve T/j E AC([-Tj, 0],0), with 0 < Tj ::::; TO, such that T/j(O) = ,( -tj), T/j( -Tj) = Y and
I:
L[T/j] < E:
+ uo(/( -tj») -
uo(y).
J
For each j E N we fix 8j > 0 so that tj = (1 + 8j )(t j - Tj). That is, we set 8j = Tj/(t j - Tj). Define ' j E AC([-tj + Tj, 0], 0) and ~j E AC([ -tj, 0], 0), respectively, by ,j(8) = ,((1 + 8j )s) and if - tj + Tj ::::; S ::::; 0, if - tj ::::; s < -t j + Tj.
115
Noting that ~j(O) = x, ~(-tj + Tj) = /,( -tj) and ~j (-t j ) = y, we observe that if j is large enough, then
1: L[~j] J
=
l:+T Ll!j] J
+
J
1:
L[17j]
J
< uo(x) - uo(!( -tj)) + TjWI
C_2T-) + J
C
+ uo(!( -tj)) - uo(y)
J
= uo(x) - uo(y) + TjWI (~ ) + c. tj - Tj
Here we have used the fact (see for instance [16, Lemma 4.4] and the proof of [16, Theorem 4.3]) that for some modulus WI, if j is large enough, then
We combine the above with
u(x, t) :S
1: L[~jl
+ U(~j( -tj ), t - t j ),
J
to get
u(x, t) < uo(x) - uo(y)
+ TjWI
C_2T) + u(y, t - tj) + J
Sending j
-+ 00,
c.
J
we see that u(x, t) < uo(x)
+ c.
Hence, we have u(x, t) :S
uo(x). We next assume that (A8)_ is satisfied. Thanks to Lemma 3.4, for each < Tj :S TO, such that 17j(O) = /'(-tj +TO), 17j(-Tj) = y and
j E N there is a curve 17j E AC([-Tj, O],D), with 0
lOT L[17j] < C + uo(!( -tj
+ TO)) - uo(y).
J
For each j E N we set 5j = (70-Tj )/(tj -Tj), so that tj -TO = (1-5 j )(tj -Tj) and 5j E (0, 1). Define /'j E AC([-tj + Tj, O],D) and ~j E AC([-tj, 0], D), respectively, by /'j(8) = /,((1 - 5j )8) and
+ Tj :S 8 :S 0, tj :S 8 < -tj + Tjo
if - t j if -
116
observe as before that for some modulus w!, if j is large enough, then
TO -
7-
< uo(x) - uo(r( -tj + 70)) + t __ / (tj J
70)W!
J
(70 - T-) t __ / J
J
+c + uo(r( -tj + 70)) - uo(y) ::; uo(x) - uo(y) Thus we get
u(x, t) ::;
1:
L[f.j]
+ 70W! (~) + c. tj -
7j
+ u(f.j( -tj), t - tj)
J
< uo(x) - uo(y) + 70W!
(t _~ T-) + u(y, t - tj) + c. J
J
Sending j ----- 00, we conclude that u(x, t) ::; uo(x).
o
Proof of Theorem 3.1. Assume first that A = 0. Then b is the unique solution of (D) and uo(x) == 00. Hence, we have u(x, t) = b(x, t) 1\ uo(x) for all (x,t) E TI x R Next we assume that A -=I- 0. Recall that u_(x) = uo(x) for all x E A, that Uo is a solution of (SC) and that b is a solution of (D). In particular, b(x, t) ::; g(x, t) for all (x, t) E an x R It is now easy to check that the function v(x, t) := uo(x) 1\ b(x, t) is a solution of (D). Furthermore, we find from (10) or (13) that
u(x, t) ::; b(x, t)
for all (x, t)
E
TI x R
(14)
According to Lemma 3.6, we have u(x, t) ::; uo(x) for all (x, t) E TI x R Hence, we see that limt-->-oo u(x, t) = uo(x) for all x E A. Since u E BUC(TI x JR.), by the Ascoli-Arzela theorem, we infer that the above convergence is uniform for x E A. We now fix any c > 0 and choose a 7 E JR. so that !u(x, t) - uo(x)! ::; c for all (x, t) E A x (-00,7]. By (14), we see that !u(x, t) - v(x, t)1 ::; c for all (x, t) E A x (-00, 7]. We apply Lemma 3.5, to observe that lu(x, t) - v(x, t)! ::; c for all TI x (-00, 7]. Moreover, we apply a comparison theorem for the initial-boundary value problem for (D) in TI x (7, 00), with initial data u(·, 7) and v(-, 7) ± c, to conclude that !u(x, t) - v(x, t)1 ::; c for all (x, t) E TI x R Finally, noting that c > 0 is 0 arbitrary, we complete the proof.
117
Proof of Theorem 3.2. We set
for x EOn,
g_ (x) := lim inf g(x, t) = inf g(x, t) tEIR
t->-oo
and note that
+ g_ (y)
inf b(x, t) = inf{ d(x, y)
tEIR
for all x E O.
: yEan}
(15)
Indeed, we see immediately that inf b(x, t) 2: inf{ e(x, y, T)
tEIR
+ g_ (y)
: yEan, T > O}
= inf{d(x,y) + g_(y) : yEan}
for all x E O.
On the other hand, for any € > 0, yEan and x E 0, there are T E lR and 0' > 0 such that g_(y) > -€ + g(y,T) and d(x,y) > -€ + e(x,y,O'). Then,
d(x,y)+g_(y) > -2€ +g(y,T) + e(x, y, 0')
2: -2€ + b(x, T + 0') 2: -2€ + inf b(x, t). tEIR
Thus, (15) holds. Next we show that liminfb(x,t) = inf{d(x,y) t~-oo
+ g_(y): yEan}
for all x E O.
(16)
for all x E O.
(17)
In view of (15) we need only to show that liminfb(x,t):S inf{d(x,y) t--+-oo
+ g_(y): yEan}
For any (x, y, €) E oxanx (0, 00), there are aT> 0 and a sequence {tj} C lR diverging to -00 such that d(x, y) > -€+e(x, y, T) and g_ (y) > -€+g(y, tj) for all j. Adding these two yields
for all j, which guarantees that (17) is valid. Now, by Theorem 3.1 and (16) we see that
u_(x) :S liminf b(x, t) = inf b(x, t) t->-oo
tEIR
for all x
E
O.
Consequently, we find by Theorem 3.1 again that for any x E A,
u(x, t) = uo(x) which was to be shown.
1\
b(x, t) = uo(x),
o
118
Acknowledgments The first author was supported in part by Grant-in-Aid for Scientific Research, No. 18204009 and 20340026, JSPS. The second author was supported in part by Grant-in-Aid for JSPS Fellows.
References 1. M. Bardi and 1. Capuzzo-Dolcetta, Optimal control and viscosity solutions
2. 3.
4.
5. 6.
7.
8. 9.
10.
11.
12. 13.
14.
15. 16.
of Hamilton-Jacobi-Bellman equations. With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications. Birkhauser Boston, Inc., Boston, MA, 1997. G. Barles, Solutions de viscosile des equations de Hamilton-Jacobi, Mathematiques & Applications (Berlin), 17, Springer-Verlag, Paris, 1994. G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton. Jacobi equations, SIAM J. Math. Anal. 31 (2000), no. 4, 925-939. E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for HamiltonJacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 15 (1990), no. 12, 1713-1742. M. Bostan and G. Namah, Time periodic viscosity solutions of HamiltonJacobi equations. Commun. Pure Appl. Anal. 6 (2007), no. 2, 389-410. M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67. A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal. 38 (2006), no. 2, 478-502. Weinan E, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math. 52 (1999), no. 7,811-828. L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math. 57 (2004), no. 4, 445-480. A. Fathi, Theoreme KAM faible et theorie de Mather pour les systemes lagrangiens. C. R. Acad. Sci. Paris Ser. I Math. 324 (1997), no. 9, 10431046. A. Fathi, Weak KAM theorem in Lagrangian dynamics, to appear. A. Fathi and A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004), no. 2, 363-388. A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185-228. N. Ichihara and H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations with semi periodic Hamiltonians, Comm. Partial Differential Equations 33 (2008), no. 4-6, 784-807. N. Ichihara and H. Ishii, The large-time behavior of solutions of HamiltonJacobi equations on the real line, Methods Appl. Anal. 15 (2008), no. 2. N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi
119
17.
18.
19.
20. 21.
22. 23.
equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal. (DOl) 10.1007/s00205-008-0170-0. H. Ishii, Homogenization of the Cauchy problem for Hamilton-Jacobi equations, Stochastic analysis, control, optimization and applications, pp. 305324, Systems Control Found. Appl., Birkhauser Boston, Boston, MA, 1999. H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) no. 2,231-266. H. Ishii and H. Mitake, Representation Formulas for Solutions of HamiltonJacobi Equations with Convex Hamiltonians, Indian Univ. Math. J. 56 (2007), no. 5, 2159-2184. H. Mitake, Asymptotic solutions of Hamilton-Jacobi equations with state constraints, Appl. Math. Optim. 58 (2008), no. 3, 393-410. H. Mitake, The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations, NoDEA Nonlinear Differential Equations Appl. 15 (2008) no. 3-4, 347-362. H. Mitake, Large time behavior of solutions of Hamilton-Jacobi equations with periodic boundary data, preprint. J.-M. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80 (2001), no. 1,85-104.
120
ASYMPTOTIC EXPANSION METHOD FOR LOCAL VOLATILITY MODELS N. ISHIMURA * Department of Mathematics, Graduate School of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan * E-mail: [email protected] www.econ.hit-u.ac.jp K. NISHIDA
Aflac Japan E-mail: [email protected] We are concerned with an asymptotic expansion method which is applied to the volatility coefficient in the Black-Scholes models. Employing the partial differential equation (PDE) approach in a weak setting, we show that the explicit asymptotic pricing formula is possible together with mathematical convergence result.
Keywords: Black-Scholes models, volatility coefficient, asymptotic expansion.
1. Introduction
In the last 30 years various contingent claims have been proposed and actively traded in financial markets, whose theoretical background are in some extent based on the renowned breakthrough made by F. Black and M. Scholes 4 as well as R.C. Merton 21. We recall that the basic Black-ScholesMerton model consists of one random security and a risk-less cash account bond, whose prices, denoted by St and B t respectively, are assumed to follow dSt = aStdWt dB t = rBt dt ,
+ JiSt dt ,
(1)
where t denotes time and W t stands for the one-dimensional Brownian motion. Fixed constants r, a, and Ji are the risk-less interest rate, the stock volatility, and the stock drift, respectively. The no-arbitrage value
121
C(St, t; K, T) of a European call option with exercise K and maturity T then satisfies the linear partial differential equation (PDE) ac 1 2 2a2c ac [jf+2 a S aS 2 +rS -rC=O, C=C(S,t) as in (S,t) E (0,00) x (O,T)
(2)
C(S,T) = max{S - K,O}. Upon solving (2) the celebrated Black-Scholes formula is derived 425. It has been reported, however, that discrepancies are observed between real markets and what the Black-Scholes depicts. One of major deficiencies stems from the assumption that the volatility a is kept fixed in (1); so-called implied volatilities usually have different values according to the variation of exercise prices 10. Much effort has been paid to remedy such situation. See 23571416 for instance and the references cited therein. One way to overcome this difficulty is to replace the constant volatility a by a function a( St, t) of the underlying asset and the time. These types of models are sometimes referred to as local volatility models 3 and they are somewhat popular among practitioners for their simplicity. The dynamics of these extended models is known to be governed by
dSt = a(St, t)StdWt dB t = rBt dt ,
+ /1Stdt,
(3)
and the corresponding PDE for the price of European call option becomes
ac [jf
1
2
2a2c
ac
+ 2a(S,t) S aS 2 +rS as - rC = 0,
C = C(S,t)
in (S,t)E (0,00) x (O,T)
(4)
C(S,T) = max{S - K,O}. The abstract existence and the uniqueness of solutions to (4) for sufficiently regular a(S, t), say uniformly Holder continuous with < ml < a(S, t) < m2 < 00 for some constants mi (i = 1, 2) is rather standard 613. In this paper we try to apply an asymptotic expansion method for local volatility models and to rigorously establish the convergence result. The strategy of asymptotic expansion itself has been already employed in the foregone literature from the different point of view. See the last paragraph of this section. The novelty of our present research is to reformulate a clear scheme within the theory of PDE, and the stress is placed on to ensure that the process really functions effectively with the convergence results of mathematical rigor.
°
122
To be specific we now introduce an independent parameter £ (1£1 and we assume that there exist expansions
+ ea1 (S, t)2 + £2 a2 (S, t)2 + ... = Co(S,t) +eC1(S,t) +£2C2(S,t) + ... ,
«
1)
a(S, t)2 = a5 C(S,t)
(5)
where a5 is a base constant. The assumption on {an(S,t)2}~=1 will be specified later (see (6)). The expressions (5) should be consistent with PDE (4). We note that the expansion of the volatility in (5) is concerned for a(S, t)2 not for a(S, t), which seems pertinent to the investigations below. Our main observation of this article states that the expansions (5) are well defined and moreover the exact pricing formula for {Cn (S, t)} ~=o is also settled in elementary way by the use of PDE theory. To proceed further we introduce some function spaces. 1
2
Eo:= {V E L1oc(0,00) IIWilEo E 1 := {V E Eo E-1
:=
IIWII~,
:=
:=
[00
Jo
IWII~o +
V(S)
2
dS
S < oo}
1 (sfJ~iS))2d: 00
< oo}
E; the dual space of E1 with respect to the inner product of Eo.
Furthermore we additionally introduce o
E1
:=
1 fJV {V E L1oc(0, (0) IS fJS E Eo}.
For VEE? we define IWIIEY := IISfJV/fJSIIEo. It is easy to see that Eo, E 1, E-1 are respectively equivalent to L2(R), H1(R), H- 1(R) via the change of variable u(x) = V(e X ) = V(S) (x = logS E R). We next clarify the hypotheses on {an(S,t)2}~=1. We assume that {an(S, t)2}~=1 are all uniformly bounded sufficiently smooth functions such that (6)
where la;loo := sUPs>O,O
Theorem 1.1. Under the assumption (6) there exist a family of functions {Cn(S, t)}~=o C LOO(O, T; Eo) n L2(0, T; E?) such that the expansions (5)
123
are well defined in consistent with (4). In particular the expansion 0/C(8, t) is absolutely convergent in L=(O, T; Eo) n L2(0, T; E~) with respect to c:
(1c:1 < 1). For example of the theorem we have
Co (8, t) = 8N(log(8/ K) + r(T - t) aovT-t
+ ao~)
_ Ke- r(T-t l N(log(8/K)
2
+ r(T -
ao~
t) _ ao~) 2
T C 1(8,t)=l dT (= G(8/R,t-T)e-r(T-t)a1(R,T)2 1 . t Jo V27ra5(T - T) 1 (log(R/K) +r(T-T) . exp [ - 2 aoVT - T
+ aoVT-T)2] dR 2
'
(7) where 1
N(d) := ,j2;
jd_= exp [ - "2] y2 dy
denotes the cumulative distribution function for a standarised normal random variable and
./ 2 T )2] 1 [ 1(IOg8+r(T-t) G (,t):= 8 V 27ra5(T-t)ex p -4 V a 5(T-t)/2 +yao( -t)/2 (8) is the Green function of the Black-Scholes PDE; that is, the kernel of the left hand side of the equation (2) with a = ao. We remark that Co (8, t) is nothing but the usual Black-Scholes formula. The other Cn (8, t) (n ~ 2) can be attained stage by stage. These Cn (n ~ 2) will be defined weakly, which is provided in Definition 2.1 of §2 below. The technique of asymptotic expansions is known to be a powerful tool in applied mathematical sciences. To mention one example we recall that it is essentially utilized to derive an interface motion in the reaction diffusion systems 8924, by which our present research is motivated. For applications in financial analysis we are unable to advance without recalling nice works by J.-P. Fouque et al. 1112, where stochastic volatility models are dealt with, which can be seen to include as a particular case our local volatility models. However, there seems to be little literature concerning the convergence results. Our current project was first undertaken by K. Nishida 22. We also refer to recent paper by N. Kunitomo and A. Takahashi 20 in which
124
the asymptotic expansion approach is taken along the context of stochastic calculus. For a relating result we refer to T. Fujita 15. The organization of the paper is as follows: In §2 we formulate our main results of this paper. §3 is devoted to Discussions.
2. Proof of Theorem
We begin with the observation that the linear function C(S) = as (a E R) solves the Black-Scholes equation (4). It is legitimate to consider the next modified equation.
ac 1 2 2a2c ac at + 20"(S, t) S aS 2 + rS as -
rC = 0,
C = C(S, t)
in (S, t) E (0, (0) x (0, T)
(9)
C(S, T) = max{S - K,O} - SEE? We wish to construct a solution C(S, t) of (9) via the expansion (5), whose behavior is prescribed by Cn(S, t) ~ exponentially as S ~ 00 for every n = 0,1,2,···. We use the same C(S,t), {Cn(S,t)}~=o with the abuse of notation.
°
Remark 2.1. Since the linear growth term S of C(S, T) = max{ S - K, O} is concerned only with Co(S, t), there is no need for this modification in a sense. We do this procedure just for the simplicity of the presentation. Now we formally place the expansion (5) into the equation (9) and investigate it according to the order of c.
0(1). We derive aGo
1
2
---at + 20"0S
2a 2Go aS 2
+ rS
aGo as - rGo =
° m. (S, t) E (0, (0) x (0, T)
Go(S, T) = max{S - K,O} - S. This is readily solved; the result is provided by the famous Black-Scholes formula with a modification of - S term.
Go(S, t)
= SN(log(S/K) + reT O"ovT - t
t)
+ o"o~) 2
_Ke-r(T-t)N(lOg(S/K)+r(T-t) _ O"O~) -So O"oVT - t 2
125
0(£). We see that
oC
I
at1 + 2"O"oS 2
2 02C1
OS2
oC1
+ rS aS - rC1 I
2
202CO
= -2"O"l(S,t) S OS2
in (S,t) E (0,00) x (O,T)
C 1 (S,T) = O. Since the Green function G(S, t) of the left hand side ofthe equation is given by (8), the solution C 1 (S, t) is expressed as in (7) thanks to the variation of constants formula. We moreover note that
(10) for some constant M which is independent of £. At this point we describe the notion of weak solutions (see also 18). To keep the generality we deal with the next nonlinear equation.
OV
I
2
&t + 2"O"oS
202V OS2
oV
+ rS aS
- rV = F(V, S, t),
v = V(S, t)
in (S,t) E (0,00) x (O,T)
(11)
V(S,T) = Vo, where F is a suitable function and Vo E Eo with Vo(O) = 0 is a given initial data.
Definition 2.1. We say V E LOO(O, T; Eo) n L2(0, T; Ed a weak solution of (11) with maturity data Vo E Eo if V(S, t) satisfies the next conditions. (WI) There holds for each W E E1 with W(O) = 0 and almost every
0:::; t :::; T that
1
oV dS -w-= oatS 00
1 {0"500
0
2
(W2) V(S, T) = Vo(S) For n
~
(ov ov ) W }dS s - - v) soW - + (-rS-+rV+F -. aS aS aS S in Eo.
2 we present a weak solution Cn(S, t) in the sense of this Definition.
126
0(10 2 ). We derive
8C2
1
2
at + 2"O"oS =
282C2 8S2
+ rS
8C2 8S - rC2
2 1 2 28 C O 1 2 2 82C1 -2"0"2(S,t) S 8S 2 - 2"O"l(S,t) S 8S 2
(12)
in (S,t) E (0,00) x (O,T) C2 (S, T)
= O.
We intend to show the existence of weak solution C 2 in the sense of Definition 2.1. To do this we deduce the a-priori estimate. Multiplying (12) by C 2 we find that
where the use of integration by parts and Schwarz inequality was made. Integrating with respect to t and invoking (6)(10) we obtain
for every 0
~
t
~
T, which implies the desired weak solution.
127
O(cn ). The step continues. We are going to solve the next equation in the above weak sense.
aC
atn + 20"0S 1
2
2
a 2c n aS2
aCn
+ rS as - rCn
( )2 2 -21 ~ ~O"k S,t S
=
k=l
Cn(S,T)
a2Cn_k aS 2
in (S,t) E (0,00) x (O,T)
= 0.
As in the case of O(c 2 ) we deduce the a-priori estimate. Multiplying the equation by Cn we infer that Id
2
12
2
2
2dt I/Cnl/ Eo (t) - 20"0 IICnl/E~ (t) - rl/Cn IIEo (t) 00 =~~ {0"2 sacn- k sacn saO"~ sacn- k c }dS 2 ~ as + as as n S k=l 0 k as
1
1
2
n
-2 (L 100~1001ICn-kllk?(t))
1/2
n
(L 100~100l/Cnl/h(t))
1/2
k=l
k=l
- ~(t IS~llooIlCn-kllk~(t)) (t IS~llooIICnl/~o(t)) 1/2
k=l
1/2
k=l
1 ~ 2 0"5 2 1 ~ I aO"~ I r 2 2 - 0"2 ~ 100k100' 4"CnIE?(t) - 2r ~ S as 00' 21ICnIlEo(t) o k=l k=l 1
a2
n
- 4 L (10"~100 + Is ; ; 1(0) I/cn-kllk?(t), k=l
1
d
2
0"5
2
r
2
2 dt IICnl/Eo(t) 2 4"Cnl/E?(t) + 2I/cnl/ Eo (t) 1
- 20"5
n
L
k=l
a2
2
(10"~100 + Is ai 1(0) (~o I/Cn-kl/k~(t)).
In light of (6) we inductively obtain
"Cnl/~o(t) + r
iT IICnll~o(T)dT
+ ~5iT "Cnl/k~(T)dT:::; M,
for every 0:::; t :::; T, from which we conclude that the asymptotic expansion for C(S, t) surely converges in Loo(O, T; Eo) n L2(0, T; E?). The proof of Theorem is thereby completed.
Remark 2.2. It is certain that we can use the variation of constants formula for (12) as in the O(c) case. Indeed we can write down the explicit
128
formula for Cn.
iT 1
00
Cn(S,t) =
dT
G(SjR,t-T)e-r(T-t).
2 ~ 2 2 a C n _k dR . ~ak(R,T) R aR2 (R,T)R' k=l
where G(S, t) is given by (8). Here we merely employ an easy way and avoid detailed estimates on the Green function. Remark 2.3. If we adopt the change of variables
x
:=
+ TT,
log S
T:=
T - t,
and the transformed prices u(x, T) := erTC(e X - rT , T - T; K, T),
then (4) is transformed into 2 au = ~a(x, T)2 (a U _ aU) aT 2 ax 2 ax u(x,O) = max{ eX - K,O}.
in (X,T) E S1 T := R x (O,T)
(13)
Since the kernel of the operator a j aT - 2 -1 a5 (a 2j ax 2 - ajax) is
1
\ i27ra 5T exp
[1 -"2 (X aoVT -
a OVT ) -2-
2] ,
we infer that the formula of un(x, T) is expressed as Un ( x, T ) -_
1
2J27ra5
lTdJoo~an(Y'17)2(a2un-i aUn-i)( ) 17 ~ - -2- - - - Y,17' 0
-00
.exp[-~(
i=l
ax
yh - 17
x-Y
2 ao';T -17
ax
_ao JT=17)2]dy. 2
These are totally equivalent to the formula in terms of the original variables. 3. Discussions
We have developed the process of asymptotic expansions applied to local volatility models. Thanks to elementary PDE machinery we make sure that the current method works well to successively induce approximating pricing formula. One novelty of our research consists of the mathematical convergence results, which seems missing in the existence literature. The obtained formula, however, tend to involve a complicated combination of parameters,
129
which seem unfortunately relevant to the exact pricing issues 11719 apart from the celebrated Black-Scholes formula. For example, if we take a simple data, so-called a buttefiy spread type, such as
(T1 (8, t)2 = max{ 8 - 1, O} - 2 max{ 8 - 2, O} (Tn (8, t)2 = 0 (n;::: 2),
+ max{ 8
- 3, O}
then the resulting expression would be cumbersome in explicit computation, as one can easily imagine even from the formula of C 1 (8, t) of (7). Nevertheless we believe that our achievements, which are principally mathematical results, are valuable even for man of affairs, partly because of the fact that they are explicit as well as accessible to implementation. It is our hope that the expansions of the volatility coefficient may approximate and recover the market data; if possible we may aim to make connection with the calibration problem. This is indeed carried out in part by the last section of 22, where some practical models are proposed. On the other hand numerical experiments are important and indispensable in the study of nonlinear sciences, which may apply to our current topics. Since our formula is in fact exact, it has a potential to be ready to be compared directly with the market data. Calibration and determination of admissible volatility models, which is accompanied with numerics, should be pursued. This theme for researches would be challenging and it will be a subject of our future study. Acknowledgments We are grateful to Professor Takahiko Fujita for warm encouragement. The work of the first author (NI) is partially supported by Grants-in-Aids for Scientific Research (No. 16540184), from the Japan Society for Promotion of Sciences, as well as by the Inamori Foundation through the Inamori Grants for the fiscal year 2006-2007. References 1. E. Barucci, S. Polidoro, and V. Vespri, Math. Models Meth. Appl. Sci. 11, 475 (2001). 2. M. Baxter and A. Rennie: Financial Calculus, (Cambridge University Press, Cambridge, 1996). 3. H. Berestycki, J. Busca, and I. Florent, Quant. Finance 2, 61 (2002). 4. F. Black and M. Scholes, J. Political Economy 81, 637 (1973). 5. I. Bouchouev and V. Isakov, Inverse Problems 13, Lll (1999). 6. 1. Bouchouev and V. Isakov, Inverse Problems 15, R95 (1999).
130
7. 1. Bouchouev, V. Isakov, and N. Valdivia, Quant. Finance 2, 257 (2002). 8. G. Caginalp and P. C. Fife, SIAM J. Appl. Math. 48, 506 (1988). 9. J. W. Cahn, C. M. Elliott, and A. Novick-Cohen, Euro. J. Appl. Math. 7, 287 (1996). 10. B. Dupire, RISK J. 7 18 (1994). 11. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility (Cambridge University Press, Cambridge, 2000). 12. J.-P. Fouque, G. Papanicolaou, K. R. Sircar, and K. Solna, SIAM J. Appl. Math. 63, 1648 (2003). 13. A. Friedman, Partial Differential Equations of Parabolic Type (Krieger Publishing Company, Florida, 1983). 14. T. Fujita, Introduction to the Stochastic Analysis for Financial Derivatives (Finance no Kakuritsu-kaiseki Nyumon) (Kodan-shya, Tokyo 2002, (in Japanese». 15. T. Fujita, Asia-Pacific Financial Markets 9, 211 (2002). 16. J. Hull, Options, Futures, and Other Derivatives, 4th edition (Prentice-Hall, New Jersey, 2000). 17. H. Imai, N. Ishimura, 1. Mottate, and M. A. Nakamura, Asia-Pacific Financial Markets 13, 315 (2007). 18. N. Ishimura, preprint (2007). 19. N. Ishimura and T. Sakaguchi, Asia-Pacific Financial Markets 11, 445 (2006). 20. N. Kunitomo and A. Takahashi, Math. Finance 11,117 (2001). 21. R. C. Merton, Bell J. Econ. Manag. Sci. 4, 141 (1973). 22. K. Nishida, Thesis for the Master-Course degree, (Graduate School of Economics, Hitotsubashi University, 2006, (in Japanese». 23. P. Protter, Stoch. Processes Appl. 91, 169 (2001). 24. J. Rubinstein, P. Sternberg, and J. Keller, SIAM J. Appl. Math. 49, 116 (1989). 25. P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives (Cambridge University Press, Cambridge, 1995).
131
RECENT DEVELOPMENTS ON MAXIMUM PRINCIPLE FOR LP-VISCOSITY SOLUTIONS OF FULLY NONLINEAR ELLIPTIC/PARABOLIC PDES SHIGEAKI KOIKE
Department of Mathematics, Saitama University, Sakura, Saitama, 338-8570 Japan [email protected] The ABP type maximum principle for LP-viscosity solutions of fully nonlinear second order elliptic/parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is exhibited.
Keywords: viscosity solution; maximum principle; strong solution.
1. Introduction
This survey article presents recent results of a series of works with Swi~ch, and some extensions by Nakagawa. We are concerned with recent developments on the AleksandrovBakelman-Pucci (ABP for short) maximum principle for fully nonlinear second order uniformly elliptic partial differential equations (PDEs for short) in a bounded domain n eRn:
F(x, Du(x), D 2 u(x)) = f(x)
in
n,
(1.1)
where F : n x R x sn --> Rand f : n --> R are given measurable functions. Here sn denotes the set of n x n symmetric matrices equipped with the standard ordering. We also discuss on the parabolic version, the Aleksandrov-BakelmanPucci-Krylov-Tso (ABPKT for short) maximum principle for fully nonlinear second order uniformly parabolic PDEs in Q := n x (0, T]: n
Ut(X,
t)
+ F(x, t, Du(x, t), D A
2
u(x, t))
= f(x, t) A
in Q,
(1.2)
where P : n x (0, T] x R n x sn --> Rand j : n x (0, T] --> R are measurable. We note that F and P do not depend on u itself here.
132
For the sake of simplicity, throughout this paper, we assume n
c B l , and T
= I,
where Br = {x E R n I IIxll < r} for r > O. It is easy to extend all the results below to general bounded domains nand T > 0 by scaling and translation. The maximum principle for solutions is the first step in the regularity theory for weak solutions of fully nonlinear elliptic/parabolic PDEs (1.1)/(1.2). To this end, as in [14,15]' we shall introduce the associated extremal PDEs. For fixed uniform ellipticity constants 0 < ,\ ::; A, we define S>:,A = {X E sn I ,\[ ::; X ::; AI}. Then, we denote by p± = Pt,A the Pucci extremal operators: for X E sn, P+(X)
= max{ -trace(AX) I A
E S>:,A},
IA
E S>:,A}'
P- (X) = min{ -trace(AX)
Note that the following inequalities hold true: P-(X)
+ P-(Y)
::; P-(X
+ Y)
::; P-(X) + P+(Y) ::; P+(X + Y) ::; P+(X)
Since we may assume that F(x, 0, 0)
t
+ P+(Y).
= F(x, t, 0, 0) = 0 for
x E nand
E (0,1] without loss of generality, we suppose that
P-(X - Y) - J.L(x)(I~lm-l ::; F(x,~,X) - F(x,1], Y) ::; P+(X - Y) + J.L(x)(I~lm-l
for x E
n,~,1] E
+ 11]lm-l + 1)1~ -1]1 + 11]lm-l + 1)1~ -1]1
Rn,X, Y E sn, and
P-(X - Y) - P(x,t)(I~lm-l ::; F(x, t,~, X) - F(x, t, 1], Y) ::; P+(X - Y) + Mx, t)(I~lm-l
+ 11]lm-l + 1)1~ -1]1 + 11]lm-l + 1)1~ -
1]1
for x E n,t E (0, 1],~,1] E Rn,X, Y E sn, where m ~ I, J.L E L~(n) and PEL ~ (Q) for 1 ::; q ::; 00. Here and later L ~ (-) is the set of nonnegative functions in Lq(-). In this paper, we consider extremal PDEs associated with (1.1) and (1.2): for given J.L E L~(n) and f E £P(n), P±(D 2 u) ± J.L(x)IDul m
and for P E L~(Q) and Ut
= f(x)
in
n,
(1.3±)
f E £P(Q),
+ P±(D2 u) ± Mx, t)IDul m =
f(x, t)
in Q
(1.4±)
133
where m ~ l. To recall the well-known maximum principle, we introduce some function spaces. We denote by W 2 ,p(n) the standard Sobolev space while W2,1,p(Q) is the set of functions u E U(Q) such that U Xi E U(Q), U XiXj E U(Q) (1 ::; i,j ::; n) and Ut E LP(Q). Also, ~!':(n) is the set of functions U : n ..... R such that U E W 2 ,p(n') for any n' <s n while WI!'~ ,p ( Q) denotes the set of functions U : Q ..... R satisfying U E W 2,1,p(Q') for any open sets Q' C Q such that distp(Q', apQ) > 0, where apQ := aQ\(n x {I}) is called the parabolic boundary of Q. Here, we use the parabolic distance distp(A, B) = inf{([x - X'[2 + [t - t'[)1/2 [ (x, t) E A,(x',t') E B} for A,B C R n x R. Theorem A. (Aleksandrov [1], Bakelman [2]) There exist constants Ck = 2n Ck(n, A, A) > 0 (k = 1,2) such that if f.L E L~(n), and U E C(n) n Wlo~ (n) satisfies
P-(D 2 u) - f.L(x) [Du[ ::; f(x)
a.e. in
n,
then it follows that sup U
::;
sup u+
+ CleC2111-'+1I2n(n) IIf+ II Ln(r[u+;n]) ,
n an where r[u+; n] denotes the upper contact set of u+;
r[u+; n] = {x E n [ 3~ E R n such that u+(y) ::; u+(x)+(~, y-x) for YEn}.
We have used the notation: f+ = max{j,O} and f- = max{ - f, O}. A parabolic version is as follows: Theorem B. (Krylov [18], Tso [21]) There exist constants C k = Ck(n,A,A) > 0 (k = 1,2) such that if P E L~+1(Q), and u E C(Q)n WI!'~ ,n+1 ( Q) satisfies Ut
+ P- (D 2 u) - jl(x, t)[Du[ ::; j(x, t) a.e.
in Q,
then it follows that +
s~pu ::; ~~gu + Cle
C211{t+II~~~'(Q)
where f'[u+; Q] = {(x, t) E Q [ 3~ (~,y - x) for y E n,o < s::; t}.
E
'+ n+l
[If
[[Ln+l(f[u+;Q])'
R n such that u+(y, s) ::; u+(x, t) +
This paper is organized as follows: In section 2, we give notations and known results from [6] etc. Section 3 is devoted to the study of elliptic
134
extremal PDEs. We present results corresponding to parabolic extremal PDEs in section 4. Although we will not go to the details of proofs, we will indicate which "strong solvability" (i. e. the existence of LP-strong subsolutions/supersolutions) is needed to cancel the inhomogeneuos terms. In sections 3 and 4, we also discuss the ABP and ABPKT type maximum principle also when PDEs have super linear terms in Du though the maximum principle fails in general at least in the elliptic case. See counter-examples in [14,15]. Acknowledgement The author would like to thank the referee for several comments and careful readings of the first draft.
2. Preliminaries First of all, we recall the definition of LP-viscosity solutions of (1.1). Whenever we consider elliptic PDEs, we suppose that n p> 2' under which u E W,!':(D) is in C(D), and twice differentiable at almost every xED; 1
u(y) = u(x) + (Du(x), y - x) + 2(D 2u(x)(y - x), y - x) + o(ly - xI 2). Definition 2.1. We call u E (resp., supersolution) of (1.1) if
C(D)
an LP-viscosity subsolution
essliminf{F(x,D¢(x),D2¢(x)) - f(x)}:::; 0 X~Xo
(res p ., ess
li~sx~p{F(x, D¢(x), D2¢(X)) -
f(x)}
20)
whenever for ¢ E W,!:(D), Xo E D is a local maximum (resp., minimum) point of u - ¢. A function u E C(D) is called an LP-viscosity solution of (1.1) if it is both an LP-viscosity subsolution and an LP-viscosity supersolution of (1.1). In order to indicate correct inequalities, we will often say LP -viscosity subsolution (resp., supersolution) of
F(x, Du, D 2u):::; (resp., 2) f(x),
135
if it is an LP-viscosity subsolution (resp., supersolution) of (1.1). We will use this kind of notations also for LP-strong sub- and supersolutions of (1.1), and (1.2) below.
Definition 2.2. We call U E C(fl) n ~!'~(fl) an LP-strong subsolution (resp., supersolution) of (1.1) if U satisfies
F(x, Du(x), D 2 u(x))::; (resp., ~) f(x)
a.e. in fl.
Whenever we consider parabolic PDEs, we suppose that
n+2
p> -2-'
under which u E Wr!~'P(Q) is in C(Q), and twice differentiable in x, and once differentiable in t at almost every (x, t) E Q;
u(y, s)
= u(x, t) + (Du(x, t), y - x) + Ut(x, t)(s - t) 1 +2(D 2 u(x, t)(y - x), y - x) + o(ly - xl 2 + Is - tJ).
Definition 2.3. We call u E C(Q) an LP-viscosity sub solution (resp., supersolution) of (1.2) if
ess
liminf
(x,t) ..... (xo,to)
{¢t(x, t)
{¢t(x,t) ( ::S:·'limsup (x,t) ..... (xo,to)
+ F(x, t, D¢(x, t), D2¢(x, t)) -
j(x, t)} ::; 0
+ F(x,t,D¢(x,t),D 2 ¢(x,t)) - j(x,t)}
~ 0)
whenever for ¢ E Wr!'c1,P(Q), (xo, to) E Q is a local maximum (resp., minimum) point of u - ¢. A function u E C(Q) is an LP-viscosity solution of (1.2) if it is both an LP-viscosity sub- and supersolution of (1.2). As in the elliptic case, we introduce the notion of LP-strong solutions of (1.2).
Definition 2.4. We call u E Wr!,;,P(Q) an LP-strong subsolution (resp., supersolution) of (1.2) if u satisfies
Ut(x, t)
, 2 + F(x, t, Du(x, t), D (x, t))::;
(resp.,~)
'
f(x, t)
a.e. in Q.
Remark 2.1. If u is an LP-viscosity subsolution (resp., supersolution) of (1.1), then it is also an Lq-viscosity subsolution (resp., supersolution) of (1.1) provided q ~ p. On the contrary, if u is an LP-strong subsolution (resp.,
136
supersolution) of (1.1), then it is also an £q-strong subsolution (resp., supersolution) of (1.1) provided p ~ q. These relations also hold for parabolic PDEs (1.2). We recall the ABP maximum principle for £P-viscosity subsolutions of (1.1) under continuity assumptions on IL and f. Although the following result is not explicitly written in [6], we can follow the argument therein to prove it. We shall only consider one of extremal PDEs (1.3±) when m = 1:
(2.1) Proposition 2.1. (cJ. [6, Proposition 2.12]) There exist Ck = Ck(n, A, A) > 0 (k = 1,2) such that if IL, f E £+.(n) n C(n), and u E C(O) is an £n-viscosity subsolution of (2.1), then it follows that supu :::; supu+
o
aO
+ CleC2111'IILn(fl) IlfIILn(qu+;o]).
Although we need to suppose that IL and f are continuous in the above, it is possible to drop the continuity for f if we establish the existence of £n-strong subsolutions of
under v = 0 on an. However, unfortunately, in order to obtain W1!;-estimates on approximate solutions, we cannot apply the argument in Lemma 3.1 in [6] since we have to suppose that IL is approximated by smooth functions in II . 1100 norm by a technical reason. Therefore, up to now, we do not know if it is possible to establish the maximum principle for £n-viscosity solutions of (2.1) only when IL E £n(n). Theorem 2.1. (cf [6, Proposition 3.3]) There exist C k = C k (n, A, A) > 0 (k = 3,4) such that if f E £+.(n), IL E £,+(n) n C(n), and u E C(O) is an £n-viscosity subsolution of (2.1), then it follows that sup u :::; sup u+
o
+ C3eC4111'1ILOC(fl) Ilf+ IILn(o).
aO
In order to explain how we use the existence of £P-strong solutions, we briefly give a proof of the above theorem. Sketch of Proof: Since IL is bounded, we may suppose IL == 1.
137
By following the argument of proof of Lemma 3.1 in [6J (see also Theorem 6.2 in [13]), there exists an Ln-strong subsolution v E C(Q) n wt!';(n) of P+(D 2 v)
+ IDvl
under v = 0 on an, where c
n.
>
:::; - f(x) - c
n
a.e. in
0, such that 0:::; -v :::; c(llfIILn(o)
Setting w = u + v, in view of P-(X + Y) :::; P- (X) verify that w is an LP-viscosity subsolution of P-(D 2 w) -IDwl :::; -c
in
+ c)
in
+ P+(Y), we easily
n.
From the definition, w cannot achieve its maximum over Q at x E we have supo w = sUPao w, which implies
n. Hence,
supu :::; supw + sup( -v) :::; supu+ + CllfIlLn(o). o o ao 0 ao For the ABP maximum principle when p < n, we will need a strong solvability result. Proposition 2.2. (cf [4, 5]) Let n satisfy the uniform exterior cone condition. Then, there exists p = pen, >., A) E [~, n) satisfying the following property: For p > p, there exists C 5 = C 5 (n, >., A,p) > 0 such that if f E £P(n), then there exists an £P-strong solution v E C(Q) n wj!':(n) of
(2.2) under v = 0 on
an
such that
11r IILP(O) each n' <s n, -C5
and for
:::;
v :::; C 5 1If+IILP(0)
in
n,
Il v IlW 2 ,p(o') :::; C 6 I1fll£P(0), where C 6 = C 6 (n,>.,A,p,dist(n',an))
> O.
We note that Winter [22] has recently obtained the global W 2 'P- estimate on LP-viscosity solutions of (2.2) though we do not need this result here. We now present parabolic versions of the above results. Theorem 2.2. (cf [15, Proposition 3.2]) There exist C k = Ck(n, >., A) > 0 (k = 7,8) such that if E L~+l(Q), P E L~+1(Q) nC(Q) and u E C(Q) is an Ln+l_viscosity subsolution of
1
Ut
+ P- (D 2 u) -
p,(x, t)IDul :::; lex, t)
in Q,
138
then it follows that s~pu < _ ~~gu
+
+ C7 eC8I1jlIl2!~,( Q ) Ilf'll £n+l(f'[u+;Q])'
A parabolic version of the strong solvability is as follows:
Proposition 2.3. ([8, Theorem 2.8]) Let n satisfy the uniform exterior cone condition. Then, there existp = p(n,A,A) E [~,n+l) satisfying the following property: For p > p, there exists 0 9 = 09(n, A, A,p) > 0 such that , 21 if f E LP (Q), then there exists an LP -strong solution v E 0 ( Q) n W1o'c ,P (Q) of
under v = 0 on opQ such that , '+ -0911r 1I£I'(Q) ::; v ::; 0911f II£P(Q) and for each Q'
c
Q with distp(Q', opQ) > 0,
I v llw 2 ,1,p(Q') where 010
in Q,
::; o1OlIill£p(Q),
= 01O(n, A, A,p, distp(Q', opQ» > O.
3. Elliptic PDEs We first study extremal PDEs (1.3-) with J.l E L~(n) and f E £P(n) in four separate cases: (1) m = 1, q ::::: p > p and q > n, (2) m = 1, q = n > p > P and f E Ln(n), (3) m > 1, q ::::: p > n, (4) m> 1, q> n ::::: p > p and p(mq - n) > nq(m - 1). In what follows, we simply write 1I·lI p for 11,11£1'(0) if there is no confusion.
3.1. Linear growth (i.e. (1) and (2» In our mximum principle below, the LP- norm of the inhomogeneous term is not taken over the associated upper contact sets unlike the ABP / ABPKT maximum principle. However, to establish the Holder continuity of LPviscosity solutions via the weak Harnack inequality in the next step of the regularity theory, this lack of information does not cause any problem. See [16] for the details. Our first result is the following:
139
Theorem 3.1. (cf. [15, Proposition 2.8, Theorem 2.9]) Assume that q ;::: P
>p
and
q
> n.
(3.1)
Then, there exist G = G(n, A, A) > 0 and G' = G'(n, A, A,p, q) > 0 such that if f E L~ (n), J1 E L ~ (n), and u E G (n) is an LP -viscosity subsolution of 2
P-(D u) - J1(x)IDul :::; f(x)
in no := {x En
1
u(x)
> supu+}, (3.2) 80.
then it follows that (a) when q ;::: p;::: nand q > n, we have
supu:::; supu+ 0.
and (b) when q
> n > p > p, we have
sup u :::; sup u+ 0.
+ GeClitLll~llfIILn(no)'
80.
M
+ G' {eC[[tLlI~ 11J111: + ~I 11J111~} Ilfl!LP(no)' k~
where N = N(n,p,q) E Nand PN ;::: n are given by
N= min {k EN/ Pk:= q Pk-!q 2: n, Po = p}. + Pk-I Remark 3.1. Here and later, for n > P > n/2, we define p* = that Pk+1 > Pk for k = 0,1, ... , N - 1.
nn!p'
Notice
Theorem 3.2. (cf. [15, Theorem 2.9]) Assume that q = n
> P > p.
Then, there exists G = G(n, A, A) > 0 such that if J1,f E L'"t(n), and u E G(O) is an LP-viscosity subsolution of (3.2), then it follows that
supu :::; supu+ 0.
+ GeClitLlI~ IIfIILn(no)'
80.
Remark 3.2. In Theorem 2.9 of [15]' we did not treat the case of q = n > > P because we dealt with the case when the P for f E LP(n) is the same as P for LP-viscosity solutions. We mentioned this case in [16].
P
Following the argument in Lemma 3.1 of [6], we obtain the following strong solvability. Proposition 3.1. (cf. [15, Proposition 2.6]) Let n satisfy the uniform exterior cone condition. Assume that one of the following conditions holds. (i) q 2: P 2: nand q > n, { (ii) q = n > P > p.
140
Then, there exists G = G(n,.x, A) > 0 such that if J.l E L~(n) satisfies supp J.l <s n, and f E LPvn(n), then there exists an LP-strong subsolution u E G(Q) n ltJ!f(n) and an LP-strong supersolution v E G(Q) n w1!':(n) , respectively, of P+(D 2 u)
+ J.l(x)IDul
such that u
=v =0
~ f(x)
and P-(D2 v) - J.l(x)IDvl ;::: f(x)
n
on an, and
-Ge clI J.
in
in
n,
n' <s n, lIu, vllw 2.p(n') ~ G'llflip,
where G' = G'(n,.x, A,p, q, 11J.lllq, dist(n', an)) >
o.
Remark 3.3. (1) Since we use Proposition 3.3 in a larger domain than n in the proof of Theorem 3.1 below, the hypothesis on J.l, supp J.l <s n, is not a restriction. (2) We will not use the local W 2 'P-estimate of these LP-strong sub- and supersolutions. Sketch of proof of Theorem 3.1: We use the same notation J.l and f for J.lXn o and f Xno' where xno is the characteristic function of no. To prove (a), we can use the same argument as in the proof of Theorem 2.2 together with Proposition 3.3 (i) when q ;::: p ;::: nand q > n; Choose v E C(B 2 ) n W 1!f(B2 ) of
P+(D 2 v)
+ J.l(x)IDvl
~ - f(x) -
€
a.e. in B 2 ,
such that v = 0 on aB2 , and 0 ~ -v ~ GeCIIJ.
P-(D 2 w) - J.l(x)IDwl ~
-€
in
n,
we see that sUPn w = sUPan w, which implies the conclusion. To show (b), we notice that we cannot use Proposition 3.3 because f E LP(n) with p < n. Thus, in order to cancel the right hand side as before, we can only use Proposition 2.2; find an LP-strong subsolution VI E C(B 2 ) n W 1!f(B2 ) of
P+(D 2 vI) ~ - f(x)
in B2
such that VI = 0 on aB 2 · Note that we have the following estimates:
o ~ -VI
~
Gllfll p in B 2 ,
and
IlvIllw2.p(n) ~ Gil flip'
141
It is easy to see that
Wl
:= u
+ Vl
is an LP-viscosity subsolution of
P-( D2w l) - JL(x)IDwll S JL(x)IDvl(X)1 =: hex)
in
no.
By Sobolev embedding, we have
IIhllp, S IIJLllq II DVl lip· S GIIJLllqllvlllw 2,p(n),
;'!q'
where p* = nn!p and Pl = If Pl :::: n (i.e. P:::: 2;!!:n; N = 1), then we can apply the case (a) to conclude the proof. If not, we choose an LP'-strong subsolution V2 E G(B1) n (B ) of
WI!':'
P+( D2v2) S -hex)
1
a.e. in Bl, 2
such that V2 = 0 on &B l' and 0 S -V2 S Gllh lip, S GIIJLllq IIfllp in B l' Again, we see that W2 := Wl + V2 is an LP'-viscosity subsolution of
P-( D2w 2) - JL(x)IDw21 S JL(x)I Dv2(X)1 =: hex)
in
n.
We have
IIhllp2 S
IIJLllqllDv21lp~
S GIIJLllq1lV211w 2 ,p, (n),
where pi = nn!~, and P2 = ~~q. If P2 :::: n (i.e. N = 2), then we can use (a) to conclude. Otherwise, we continue to proceed this argument until PN :::: n, where N is given in the statement. 0 Idea of proof of Theorem 3.2. We follow the same argument as in the proof for (a) in the above together with (ii) of Proposition 3.3. 0
3.2. Superlinear growth (i.e. (3) and (4» We remark that when m > 1, there are examples where the ABP maximum principle fails in general (see [14,15]) for
P- (D 2u) - JL(x)IDul m S f(x)
in
n,
(3.3)
even if JL, f E LOO(n). Therefore, we need some restriction for the maximum principle to hold. Recently, we have given a simpler proof of the ABP maximum principle for (3.3) in [17] using a new strong solvability resut for (l.3±) when m > l. However, our proofs below come from [15]. Theorem 3.3. ([15, Theorem 2.11]) Assume that q :::: P > nand m > 1. Then, there exist 8 = 8(n, >., A, m,p) > 0 and G = G(n, >., A, m,p) > 0 such that if f E L~ (n) and JL E L~ (n) satisfying
IIfll;-lIIJLllp < 8,
(3.4)
142
and
U
E c(n) is an LP-viscosity subsolution of (3.3), then it follows that
supu:::; supu+
n
an
+ C(llfll p + Ilfll;;'llpllp).
Remark 3.4. (1) Although we suppose p E Lq(n) for q ?: p, we only need to suppose p E LP(n). (2) As in Theorem 3.1, we can replace 0. by no := {x E 0. I u(x) > sUPan u+} but we will not state this fact in what follows. (3) Notice that by (3.4), we may conclude supn u :::; sUPan u+ + CllfllpSketch of proof: For simplicity of presentation, we suppose that the global W 2 'P-estimate holds since it has been shown by Winter when an E CI,I. However, if we consider larger domains with local W 2 'P-estimates, we do not need to suppose an E CI,I. Let v E C(n) n W 2 ,p(n) be an LP-strong subsolution of
P+(D2VI) :::; - f(x) such that
VI
=
in 0.
°on an with estimates
°: :; -VI:::; Cllfll p
in 0.,
and
By Sobolev embedding, we find C s >
Ilvl II W
2 ,p(n)
:::;
°such that
Cllfll p'
IIDvIiloo :::; Csllfll p, Noting that (a + b)m :::; 2m-l(a m + bm ) for a, b ?: 0, we verify that u
+ VI
WI :=
is an LP-viscosity subsolution of
P-( D2W I) - 2m-Ip(x)IDwllm :::; 2m- I p(x)I DvI(X)lm :::; 2m- IC;nllfll;;'p(x) in n. Thus, we find V2 E C(n) n W 2,p(n) satisfying that P+(D2v2) :::; -(2m- IC;n + l)llfll;;'p(x) - c a.e. in 0., where c ).
°is arbitrary. Note that IIDv21100 :::; O(llfll;;'llpllp + c) for some
0>0. It is easy to see that W2 :=
WI
+ V2
is an LP-viscosity subsolution of
P-(D2W2) - 22(m-l)p(x)IDw2Im :::; p(x){2 2(m-I)I Dv 2(x)l m - IIfll;;'} - c. Since { ... } in the above can be estimated from above by
em(llfll;;'llpllp if we suppose Ilfll;;'-lllpllp
+ c)m
- Ilfll;;'
for some
e > 0,
< 8, then the above is estimated from above by
em (811fllp
+ c)m - IIfll;;'.
143
°
Thus, for 8 < 6- 1 , we can choose small e > to make this term nonpositive. Hence, for such 8 > 0, if Ilfll;;'-1I1JLllp < 8, then we have supwz = supwz,
n
an
o
which implies the assertion. In the next theorem, we use the notation: aO
k-1
= 0,
and
ak
.
mk-1
= ~mJ = - - - (k L
m-1
j=O
~
1).
We only state the remaining result without proof because we can combine the idea of proof of Theorem 3.3 together with the iterated comparison function method in the proof of Theorem 3.1. Theorem 3.4. ([15, Theorem 2.12]) Assume that q > n > p > m> 1 satisfy p(mq - n) > nq(m - 1). Let N
= N(n,p, q, m)
P and (3.5)
be the integer given by
. {k NI Pk:= Pk-1 * Pk-1q > n, + mq
mIll
E
Then, there exist 8 = 8(n, A, A, m,p) > 0 and G = G(n, A, A, m,p) > Osuch that if f E L~(n) and JL E L~(n) satisfy
(3.6) and u E G(Q) is an LP-viscosity subsolution of (3.3), then it follows that N+1
sup U n
:::;
sup u+ an
+ G L IIJLII~k Ilfll;;,k. k=O
Remark 3.5. (1) Note that (3.5) is equivalent to q{n - m(n - p)} > pn. Thus, if n > m(n - p), then for large q > n, (3.5) holds true. For instance, because of p > ~, we may apply Theorem 3.4 to the important case when m = 2 provided q > Z:~n. (2) By (3.6) together with the definition of ak, we observe that the second term of the right hand side of the conclusion can be estimated by Gil flip·
144
3.3. Linear and superlinear growth In this subsection, we show the ABP maximum principle when extremal PDEs have linear and superlinear growth terms in Du:
P-(D 2 u) - f.LI(x)IDul- f.Lm(x)IDul m :'S /(x)
in D,
(3.7)
where f.LI E L~(D), f.Lm E Lt~(D) and / E L~(D). Since we have a super linear term, we need some smallness assumptions on f.LI and/or f.Lm. If we follow the argument in the previous subsections, it is necessary to suppose that f.LI is smalL This is not a good hypothesis because even if f.Lm == 0, we need smallness of f.LI. Nakagawa [19J has recently shown that the maximum principle holds only when f.Lm or / is small. In the next theorem, we use the constant C 2 from Proposition 2.1, and the notation:
Theorem 3.5. (cf. [19, Theorem 3.4]) Assume that qm :::: ql :::: p > n and m > 1. For f.LI E L~ (D), there exist 5 = 5( n, A, A, p, ql, lIf.Llllql) > 0,
°
C=
C(n,A,A) > and C' = C'(n,A,A,m,p,qm, f E L~(D), f.Lm E Lt=(D),
IIf.Llllql) >
° such that if
Ilfll;,-IIIf.Lmllp < 5, and u E C(IT) is an LP-viscosity subsolution of (3.7), then it follows that supu :'S supu+
n
an
+ fJ (Cllflln + C' fJmllfll;'llf.Lmllp) .
Remark 3.6. (1) As in Theorem 3.3, we only need IIf.Lmllp, and we can change the second term of the right hand side by C'II/IIp(2) The dependence of 5 on IIf.Llllql is bounded when IIf.Llllql varies in the interval [0, RJ for a fixed R > 0. We will not repeat this remark below. Sketch of proof of Theorem 3.5: We follow the same argument as in the proof of Theorem 3.3 with an LP-strong solution VI of
P+(D 2 vd such that
VI
=
°on [}B
2
+ f.LI(x)IDvII :'S -
f(x)
in B 2 ,
with W 2 ,P(D) estimates instead of VI there.
0
While we suppose that qm = ql > n in [19], we suppose here that ql > n. Because of this generalization, unfortunately, we have to give up precise estimates in [19J for the next theorem.
qm ::::
145
Theorem 3.6. (cf [19, Theorem 3.5]) Assume that qm :::: ql > n :::: p > p and m > 1. Let M = M(n,p, ql) and N = N(n,p, qm, m) be integers given by
and . N = mIll
{k NIPk:= Pk-lPk-lqm > n,po + mqm E
*
= p} .
For ttl E L~(D), there exist 6 = 6(n, A, A, m,p,ql,qm, Ilttlllql) > 0 and C = C(n,A,A,m,p,ql,qm, IIttlllql) > 0 such that if f E L~(D), ttm E L~=(D), p(mqm - n) > nqm(m - 1),
and u
E
C(!1) is an LP-viscosity subsolution of (3.7), then it follows that
Remark 3.7. (1) When qm > ql, we do not know if N > M in general. This is the reason why we could only show a rough dependence on ttl' (2) As before, we can replace the above estimate simply by Cllfll p ' To deal with the case when n ?: p > p, instead of Proposition 3.3, we need strong solvability results for (1.3+). This has been done implicitely in [15] by the ABP maximum principle (Theorem 3.3) with local W 2 ,p_ estimates as mentioned in Remark 2.10 in [15J under assumption supp tt E D. Moreover, in [16], establishing the weak Harnack inequality, we do not need to suppose that supp tt E D to get the strong solvability (also for p> n).
Proposition 3.2. (cf [16, Theorem 7.1]) Let D satisfy the uniform exterior cone condition. Assume that ql > nand n ?: p > p. Then, there exists C = C(n,A,A,p,ql) > 0 such that if f E LP(D) and ttl E L~(D), then there exists an LP-strong subsolution v E C(!1) n W1!':(D) of
146
under v = 0 on &0 with estimates -C {
bllJLll1~ + ~1 IIJLll1~l } IIf-lip
~v
~ C {bIlJLll1~ + ~1 [[JLIiI~l } [[f+[[p
in 0,
where M = M (n, p, ql) E N is from Theorem 3.6, and for each 0' <S" 0,
where C' = C'(n, A, A,p, ql, IIJLl[[qll dist(O', (0)) > O. 4. Parabolic PDEs We shall delete ~ from fl and j in the introduction. Since we have not finished all the corresponding results to the elliptic case, we will only present results in [15,19]. We recall p = p(n, A, A) E [n!2, n + 1) which is the "parabolic" constant which gives the range of exponents for the strong solvability as in Proporition 2.3. We study one of (1.4±) as in section 3. We will present our results in several cases: (1) q:::: l,q = oo,p > p, (2) m = 1, q :::: p :::: n + 2 and q > n + 2, (3) m = l,q = n + 2> p > p and f E Ln+l(Q),
1
(4)m>l,q::::p>n+2, (5) m> l,q > n + 2:::: p >
p and mp>
(m - 1)(n + 2).
We assume the constant p* defined by *
P :=
p(n + 2) n+2-p
for p < n
+ 2.
We recall the anisotropic Sobolev embedding.
Proposition 4.1. (e.g. [8]) If p > n + 2, then W 2 ,1,p(Q) c W1,Q,00(Q), where W1,Q,00(Q) = {u E LOO(Q) [ :3C > 0 such that [u(x,t) - u(y,t)[ ~ C[x - y[ forx,y E 0,0 ~ t ~ I}.
147
4.1. Bounded coefficients (i.e. (1» We first show the maximum principle for LP-viscosity sub solutions of Ut
+ P- (D 2 u) - p(x, t) IDul m
:::;
f(x, t) in Q,
(4.1)
when p is bounded, and m ~ 1. We notice that when m > 1 in the elliptic case, we need some smallness of Ilfllp or Ilpllq. Otherwise, we know counterexamples where the maximum principle fails. However, in the parabolic case, if p E L=(Q), then we do not need to suppose any smallness of p and f for the maximum principle. Theorem 4.1. ([15, Theorem 3.7]) Assume that p > n + 2 and m ~ 1. Then, there exists C = C(n,.x, A,p, m) > 0 such that if f E L~(D), p E L't(D), and u E C(Q) is an LP-viscosity subsolution of (4.1), then it follows that
supu:::; supu + c(llfll p + Ilpll=llfll;'). Q
GpQ
Sketch of Proof: By Proposition 2.3, we find v E C(Q) n Wl~'cl'P(Q) such that Vt
+ P+(D 2 v) :::; - f(x, t)
a.e. in Ql :=
B2
x (-1,1]
under v = 0 on OpQl with estimates
0:::; -v :::; Cllfll p in Q,
IIDvIIL=(Q):::; Cllfll p·
and
For the second estimate, we used Proposition 4.1. We verify that w = u + v - {2m-lcmllpll=llfll; LP-viscosity subsolution of
+ e}t
(\Ie> 0) is an
Hence, the definition yields supw Q
= supw, GpQ
which implies the assertion in the limit as e ---> o. 0 Using the above idea together with the iterated comparison function method, we can show the maximum princple for n + 2 ~ p > jj. Theorem 4.2. ([15, Theorem 3.8]) Assume that n+2 Let L = L( n, p, m) be the integer given by
min{kENlpk:=P~l
~ p
>n+2, PO=p}
> jj and m
~ l.
148
=
Then, there exists C L+'(O),
C(n,A,A,p,m)
> 0 such that if f
mp> (m - l)(n and
U
E L~(O), JL E
+ 2),
(4.2)
E C(Q) is an LP-viscosity subsolution of (4.1), then it follows that supu ::; supu + C (lIfll; Q
opQ
t IIJLII~ +
L
1
IIJLII: + Ilfll;2) .
k=O
Remark 4.1. If 1 ::; m ::; 2, then (4.2) is automatically satisfied because p
> p->
!tll 2
> -
(m-l)(n+2) m
More precisely, if n
.
+ 2 > p,
then we may take m < n~!:p.
4.2. Linear growth (i.e. (2) and (3» We next study the case when m = 1 with unbounded JL.
Theorem 4.3. ([15, Proposition 3.6]) Assume that q ~ p > n + 2 and q > n + 2. Then, there exists C = C(n,A,A) > 0 such that if JL E L~(Q), f E L~(Q), and an LP-viscosity subsolution of Ut
+P-(D2 u) - JL(x,t)IDul::; f(x,t)
in Q,
(4.3)
then it follows that
supu ::; supu + CeC[[JL[I~ti Ilflln+!. Q
opQ
Sketch of proof: We use Proposition 2.3 in a larger domain to cancel Proposition 4.1 to get LOG-estimate of Du.
f, and D
This theorem together with the iterated comparison function method implies the following:
Theorem 4.4. ([15, Theorem 3.10]) Assume that q > n
+ 2 > p > p.
Let
M = M(n,p, q) be the integer given by
min{kENlpk:=
:k-lq Pk-l
+q
~2+n, PO=p}.
Then, there exists C = C(n,A,A) > 0 such that if JL E L~(Q), f E L~(Q), and an LP -viscosity subsolution of (4.3), then it follows that
sup u ::; sup u Q
opQ
{
M-l}
+ C eC[IJLII~ti IIJLII~ + L IIJLII~ k=O
IIfll p ·
149
4.3. Superlinear growth (i.e. (4) and (5» We study the case of m > 1. Theorem 4.5. ([15, Theorem 3.11]) Assume that q ;:: p > n+ 2 and m > 1. Then, there exist ~ = ~(n, A, A,m,p) > 0 and G = G(n, A, A, m,p) > 0 such that if f E LP(Q) and IL E L~(Q) satisfy
Ilfll;-lIlILllp < ~,
(4.4)
and u E G(Q) is an LP-viscosity subsolution of (4.1), then it follows that
Remark 4.2. As in the elliptic case, only IIILllp appears in the statement, and we may replace the second term of the assertion by Gllfllp by (4.4). Theorem 4.6. ([15, Theorem 3.12]) Assume that q > n m > 1. Let N = N(n,p, q, m) be the integer given by min{kEN/Pk:= * p*q Pk-l
+ mq
+ 2 ;:: p > p and
>n+2, po=p}.
Then, there exist ~ = ~(n, A, A, m,p, q) > 0 and G = G(n, A, A, m,p, q) > 0 such that if f E L~(Q), IL E Lq(Q) satisfy
p(mq-n-2) > (m-1)q(n+2),
(4.5)
and u E G(Q) is an LP-viscosity subsolution of (4.5), then it follows that N+l
supu ::; supu+ Q
8p Q
+ G L IIILII~k Ilfll;k. k=O
Remark 4.3. (1) If 1 < m < 2 - n+2, then (4.5) is automatically satisfied. q Moreover, since (4.5) is equivalent to q{mp - (m - l)(n + 2)} > p(n + 2), if P > (m-l~n+2) > 0, then for large q, (4.5) holds. For instance, when m = 2, (4.6) holds provided q > 2:~{~~}2}' (2) As before, we can change the second term of the above by
Gllfllp"
150
4.4. Linear and superlinear growth We shall consider the following:
In this section, we set
The next theorem shows that if /-lm E LOO(Q), then we do not need any smallness assumption on /-lm.
Theorem 4.7. (cf. [19, Theorem 4.4]) Assume that p > n + 2 and m ~ 1. For /-ll E Lql(Q), there exists C = C(n,A,A,p,m, II/-lIllql) > 0 such that if f E L~(Q), /-lm E L'+(Q), and U E C(Q) is an LP-viscosity solution of (4.6), then it follows that supu Q
:s: supu + cb (1Iflln+1 + bm-Ill/-lmlloollfll;) . opQ
We next extend Theorem 4.7 to the case p E (PI, n
+ 2J.
Theorem 4.8. (cf. [19, Theorem 4.5]) Assume that n + 2 ~ p > j5 and = M(n,p,qr) and N = N(n,p,qm,m) be the integers given by
m> 1. Let M
M=min{kENlrk:= :k-Iql >n+2, ro=p}, r k _ 1 + ql
N=min{kENlpk:=
*Pk-Iqm >n+2, po=p}. Pk-I +mqm
Then, there exists C = C(n,A,A,p,ql,m,II/-lIllql) > 0 such that if f E L~(Q), /-ll E L~(Q), /-lm E L'+(Q), pm> (m -1)(n+2), and
U
E C(Q) is an LP-viscosity solution of (4.6), then it follows that N
supu:S: supu + C2)I/-lmll~-1 IIfll;k-l Q
opQ
k=1
+ cbmll/-lm[[~+l IIfll;N+l.
+ Cb[[/-lm[[~[[fll;N
151
We shall treat the case of qm <
00.
In this case, we need to assume that
f or J.Lm is small.
Theorem 4.9. (cf. [19, Theorem 4.6]) Assume that qm ?: ql ?: p > n + 2. For J.Ll E L'!:(Q), there exist t5 = t5(n,A,A,p,ql,qm,m,lIJ.Llllql) > 0, C = C(n,A,A,p,ql,qm,m,llJ.LlllqJ > 0 such that if f E L~(Q), J.Lm E L~=(Q) satisfy
and
U E
C(Q) is an LP-viscosity subsolution of (4.6), then it follows that
Remark 4.4. Again, we only need lIJ.Lmlip for the restriction, and we can replace the second term of the right hand side by Cllfll p . Theorem 4.10. (cf. [19, Theorem 4.7]) Assume that qm ?: ql > n + 2 ?: p > p. Let M = M(n,p,ql) and N = N(n,p,qm,m) be the integers given by
N=min{kEN!Pk:=
Pk-lqm >n+2, po=p}. p* +mqm
For J.Ll E L'!:(Q), there exist t5 = t5(n,A,A,m,p,ql,qm, 1[J.Llllq!) > 0 and C = C(n, A, A, m,p, ql, qm, 1IJ.L1 [[q!) > 0 such that if f E L~(Q) and J.Lm E L~(Q) satisfy
p(mq - n - 2) and
U
> (m - l)q(n + 2),
E C(Q) is an LP-subsolution of (4.6), then it follows that N+2
sUPU:::; supu+ Q
8p Q
+C
L
[[J.Lmll~':-!lIfll;k-l.
k=l
Remark 4.5. The second term can be replaced by C[[f[[p as before.
152
References 1. Aleksandrov, A. D., Majorization of solutions of second-order linear equations, Vestnik Lenningrad Univ., 21 (1966), 5-25, English Translation in Amer. Math. Soc. Transl., 68 (1968), 120-143. 2. Bakelman, I. Y., Theory of quasilinear elliptic equations, Siberian Math. J., 2 (1961),179-186. 3. Cabre, X., On the Alexandroff-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. 4. Caffarelli, L. A., Interior a priori estimates for solutions of fully non-linear equations, Ann. Math. 130 (1989), 189-213. 5. Caffarelli, L. A. and X. Cabre, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, 1995. 6. Caffarelli, L. A., M. G. Crandall, M. Kocan, and A. Swi~ch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), 365-397. 7. Crandall, M. G., K. Fok, M. Kocan, and A. Swi~ch, Remarks on nonlinear uniformly parabolic equations. Indiana Univ.Math. J. 47 (1998), no. 4,12931326. 8. Crandall, M. G., M. Kocan, and A. Swi~ch, LP- Theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. 9. Crandall, M. G. and A. Swi~ch, A note on generalized maximum principles for elliptic and parabolic PDE, Evolution equations, 121-127, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003. 10. Escauriaza, L., W 2 ,n a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J. 42 (1993), 413-423. 11. Fok, P., Some maximum principles and continuity estimates for fully nonlinear elliptic equations of second order, Ph.D. Thesis, UCSB, 1996. 12. Gilbarg, D. and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983. 13. Koike, S., A Beginer's Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13 in 2004. 14. Koike, S. and A. Swi~ch, Maximum principle and existance of LP-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms, Nonlinear Differrential Equations Appl., 11 (4) (2004) 491509. 15. Koike, S. and A. Swi~ch, Maximum principle for fully nonlinear equations via the iteated comparison function method, Math. Ann., 339 (2007), 461-484. 16. Koike, S. and A. Swi~ch, Weak Harnack inequality for LP-viscosity solutions of fully nonlinear uniformly elliptic partial differential equations with unbounded ingredients, submitted. 17. Koike, S. and A. Swi~ch, Existence for strong solutions of Pucci equations with superlinear growth in Du, submitted. 18. Krylov, N. V., Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Siberian Math. J., 17 (1976), 226-236.
153 19. Nakagawa, K., Maximum principle for LP-viscosity solutions of fully nonlinear equations with unbounded ingredients and superlinear growth terms, to appear in Adv. Math Sci. Appl. 20. Sirakov, B., Solvability of fully nonlinear elliptic equations with natural growth and unbounded coefficients, preprint. 21. Tso, K., On an Aleksandrov-Bakelman type maximum principle for secondorder parabolic equations, Comm. Partial Differential Equations, 10 (1985), no. 5, 543-553. 22. Winter, N., W 2 ,p and W1'P-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, to appear in J. Anal. Appl.
154
MULTISCALE MODELING OF ELECTRICAL ACTIVITIES OF THE HEART Chu-Pin Loh, Hui-Chun Tien 1 , Daniel Lee 2 , Chih-Hung Chang3 1 Department
of Applied Mathematics, Providence University, Taichung, Taiwan of Mathematics, Tunghai University, Taichung, Taiwan 3 Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 2 Department
In this article, we will review various important topics in the heart modeling, especially in the electrophysiology aspect. The theme includes: modeling process for three different scales (cellular scale, tissue scale, whole heart scale), numerical methods, applications of computer simulation on heart arrhythmia, and challenges in the heart modeling.
1. Introduction
Heart is an important organ of the circulation system to pump blood over whole body. The working mechanisms consist of at least electrophysiology, muscle mechanics, and blood fluid dynamics (or hemodynamics). Due to the electrical conduction and stimulations, the cardiac muscle then has the contraction and relaxation to generate force which then pumps blood to whole body. Although the anatomy of heart is very complicated, it still functions properly and is highly adaptive due to the synergistic interplay between these underlying mechanisms. To explore the functionality and/or anatomy of heart, various multiscale viewpoints have been used: molecular, cellular, tissue, whole organ and the usual probe tools are in vivo or in vitro experiments and computer simulations. Both approaches are mutually beneficial. By means of in vivo or in vitro experiments, good mathematical models of computation can then be built up. On the other hand, computer simulation can make prediction and point out the right direction to save the experimental cost. The performance disorder of heart will induce many ·Corresponding author: Department of Applied Mathematics, Providence University, 200 Chung-chi road, Shalu, Taichung County, Taiwan. Tel:+886-4-26328001 ext. 15100; Fax: +886-4-26324653; E-mail: [email protected]
155
fatal diseases, e.g. arrhythmia, stroke ... Therefore heart research is valuable. In this article, we will review the progress of mathematical modeling of electrical activities of heart. 2. Cellular or subcellular level modeling (microscopic): Ordinary differential equations The cardiac cell (myocyte) is immersed in the environment full of charged ions, e.g. Na+, Ca2+, K+, Cl-. The cell membrane has many different residing ion channels which allow specific ions to pass through selective channels. These ions are driven by the electrical field and the chemical diffusion due to the concentration gradient and the main governing equation is the Nernst-Planck equation (see [28]). By modeling the cell membrane as a electrical circuit consisting of resistor, capacitance ... as done in the pioneer work of Hodgkin-Huxley model (HHM) for squid giant neuron axon (see [22]), the governing ordinary differential equations can be formulated (see [36]). For the ion currents formula, two most common forms are the linear Ohm formula and nonlinear Goldman-Hodgkin-Katz (GHK) formula, both of which can be derived from the Nernst-Planck equation (see [28]). One important regulatory factor of the ionic currents is the gating factor. From the experiment, it was found that the channels are not always open. There are many different gating mechanisms for the residing channels such as voltage-gated, ligand-gated, ATP-gated ... For the voltage-gated channels, gating is a stochastic process. The traditional Hodgkin-Huxley approach is an empirical method to describe such gating process. For the detailed modeling process of HHM, including the experiment design, the choice of mathematical formula and parameters from the linear or nonlinear thermodynamics principle ... , please refer to [23,24]. Another more complicated gating formulation is the Markovian process. In fact, the determinants of gating phenomenon belong to the molecule scale and it is granted that the stochastic or probabilistic method is a more correct viewpoint to treat them. The solution of HHM is in fact a low dimensional manifold of the Markovian process dynamics (see [28, 53]). Note that the regulatory factors of ion channel gating are usually related to very complicated signaling pathway (see [21,51]), e.g. second messenger signaling pathway as in the neurotransmitter or hormone and so a purpose-oriented and computational efficient models are more practical. For the cardiac myocyte, specific models for the atrium ([9, 44]), ventricle ([25, 26,36,37,55,59,60,71]), Purkinje fiber ([10]), SA node ([50, 70]), pulmonary veins sleeves ([35]) have been built up. A repository of the mod-
156
Fig. 1.
Components of single cardiac myocyte modeling (adapted from [56])
Fig. 2.
Spatial coupling of cardiac myocytes (adapted from [56]).
els collection is the following website: http://www.cellml.org/index...html. The components of these models include various ionic currents on the cell membrane (sarcolemma) for potassium, calcium, sodium, chloride, Na+K+ ATP pump, Na+ -Ca2+ exchanger, Ca2+ pump ... For the intracellular part, the modeling of Ca2+ buffers (e.g. calmodulin, troponine) and the Ca2+ storage compartment, sarcoplasmic reticulum (SR), are very important, since Ca2+ plays a key role in the cardiac functions. There exist some currents which can be induced only under pathology conditions such as IK,ATP ([56]) in ischemia condition, IK,Ach when adding the parasympathetic neurotransmitter acetylcholine ... (see Figure 1,2) For more detailed description of the isolated cardiac myocyte modeling, please refer to [41].
157
3. Tissue level modeling (mesoscopic) To model the electrical activities of heart, a spatial-temporal formulation is necessary. Since the complicated anatomy and multiscale nature of heart, . a computationally efficient continuous model of average scale, bidomain model, has been derived using the homogenization technique (see [8, 28]). This model is of elliptic-parabolic type and the reaction term is the various ionic currents described previously. The realistic heart tissue can be divided into intracellular and extracellular subdomains. Under this average scale, it allows us to have a zoom-out looking at the heart tissue and assume the simultaneous existence of extra and intra domains (fractional occupation) at each space point. By assuming both domains have the same anisotropic ratio, i.e., the conductivity tensors of both domains are proportional or assuming the extracellular domain is resistance free, the more simple monodomain model can be derived which is a reaction-diffusion type model:
where V is the transmembrane potential (intracellular potential minus extracellular potential), Y = (Y1, ... , Yn) is the gating variables, S = (Sl' ... sm) is the ionic concentrations, B = (b 1 , •.• , bl) is the buffers of Ca2+, em is the surface capacitance per unit area, lion is the total ionic currents per unit area, X is the ratio of membrane area per tissue volume, Di is the intracellular conductivity tentor, and lstim is the stimulation current. When the monodomain or bidomain models incorporate more advanced formulation of ionic currents (reaction term lion), usually the total equations number is over 20 (e.g. Luo-Rudy type model) or even 100 and so alternative models of low computational load are necessary. Simplified mathematical models of activator-inhibitor type of partial differential equations are very useful to catch the dynamic insight, for example the Karma model (two components, [27]), Alieve-Panfilov model (two components, [1]), Fenton model (3 components, [14]). Another choice is the discrete type of cellular automata (CA) model. In CA model, the tissue is composed of discrete cells (or grid points). A finite number of states will be experienced for each grid point and the state change is determined by its neighbors by some
158
simple rules. In the Wiener-Rosenblueth CA model ([69]), 3 states were adapted: resting, excited, and refractory. Although possessing the advantage of low computational demanding, the too simple rules of CA models have been criticized. In the J.M.Greenberg and S.P.Hastings model ([18]), a rigorously derived CA model with theoretical basis was built up. In the recent more advanced CA models ([7, 12, 13, 17, 38, 65]), some important effects have been incorporated, such as curvature effects, dispersion relation, cell density heterogeneity, anisotropy effects. In particular, in the Chernyak-Feldman-Cohen models ([7, 12, 13]), systematic algorithms to choose proper parameters were developed and many important phenomena observed in the continuous models can also be mimicked via their CA models. Recently, C. D. Werner et al. ([68]) used the CA model to simulate whole heart excitation conduction. Also, M. Reumann et al. used CA model to explore the atrial fibrillation (AF) ([45, 46]). 4. Whole heart modeling (macroscopic) For the whole heart modeling, the main governing equations are also the ones mentioned in the tissue scale but there are some other topics necessary to deal with.
4.1. Buildup of geometric model of heart Both governing equations and the corresponding space domain are equally important for heart simulation. Since heart anatomy is very complicated (see Figure 3), the conduction anisotropic effect due to fibers direction, cell density heterogeneity (space inhomogeneous),... should be considered. To construct an anatomically correct geometric model, first we need to obtain the medical image data by CT, MRI, or ultrasound. Such images rersource can be found in the following websites: The Visible Human Project, National Library of Medicine, USA, http:j jwww.nlm.nih.gov jresearchjvisiblejvisibleJlUman.html and Society for Cardiovascular Magnetic Resonance (SCMR), http:j jwww.scmr.orgj. Using digital image processing methods, e.g. segmentation, region growing ... and some smoothing out methods, the geometric model can then be constructed (see [15, 48, 52, 54]). Notably in [42], the simultaneous construction algorithm for both geometric model and computational mesh of finite element numerical method has been established. As indicated in the Figure 3, there exists a fiber-sheet structure of the heart anatomy. Let alex) be the direction parallel to the local fiber, at(x)
159
Collagen
Fig. 3. Fiber-sheet structure of heart (adapted from B.H.Smaili et a\. [57] and I. J. LeGrice et al [32])
be the direction lying on the sheet and orthogonal to fiber, and av(x) be the normal direction of the sheet. Then the anatomical information can be incorporated into the bidomain equation as follows (see [8]):
where H is the heart region, U e is the extracellular potential, I~pp is the applied current at extracellular domain, De, Di are the extracellular and intracellular conductivity tensors, respectively and Di,e(x) = 17;,e al (x)aT(x) + 17;,e at (x)a[(x) +17~eav(x)a~ (x), for axisymmetric anisotropic media l7~e 17;,e, Di,e(x) 17;,e I + (17;,e 17;,e)al(x)aT (x), where 17;,e, 17;,e, l7~e are the conductivity coefficients along the corresponding directions al(x), at(x), av(x).
4.2. Numerical methods It takes much time to simulate the electrical activities of heart, roughly over 5 hours CPU time for one beat. Therefore good numerical schemes
160
and high performance environment are necessary. We compare briefly some numerical methods used for the simulation of cardiac electriphysiology as belows: (1). Classical finite difference method (see [49]): Although its mesh can be easily constructed, this method is valid for regular solution domain (not suitable for organ with complex anatomy like heart). (2). Generalized finite difference (GFD) (see [61]): Its features are as follows: (i). using arbitrary and irregular meshes without any need to specify element basis functions (ii). meshes can be constructed using existing ones (iii). employing an innovative approach to enforce boundary conditions. (3). Finite element method (see http://nrcam.uchc.edu/dropbox/ Continuity /Usyk-McC_computationaLmechanics. pdf or [42]): This method is a very popular one and has been adopted by many heart researchers. It is useful for complex geometric domain and can easily deal with the no flux boundary condition, especially for the heart. But it is a massive work to build up the mesh and it is computationally demanding to evaluate the integral terms. (4). Finite volume method (see [20]): This method is useful for complex deforming geometric domain as in finite element method. Furthermore, its nodes number is reduced compared to finite element method. (5). Hybrid methods: The finite element-derived finite difference method can be found in [6]. The finite element-derived hybrid method of finite Volume and finite difference can be found in [62]. (6). Pseudospectral (Chebyshev) method (see [40, 72]): . Advantage: (i). High accuracy (a). limiting accuracy version of finite difference method;
161
(b). the smoother the genuine solution, the higher the convergence rate; (c). Clustered grid points toward the boundary (usually N'(-2) spacing near the boundary, N'(-l) otherwise), which implies the reduced nodes number and reduced number of discretized equations; (ii). Unifying form to express boundary conditions . . Shortcoming: (i). not suitable for complex geometric domain; (ii). approximating solution using global nodes, which increases the involved unknown variables in every discretized equation; (iii). not suitable for variable resolution requirements in different parts of a large domain. Some other acceleration methods include the operator splitting and adaptive step method (see [43]). Note that since the huge computation load in heart simulation, the parallel computation using multiple CPUs is needed. A typical method for parallel computation is using the domain decomposition method and MPI interface to exchange informations between different subdomains ([5, 58]).
4.3. EGG computing ECG is a very useful clinical tool for the diagnosis of cardiac electrophysiology. Some formulas of ECG computation have been derived under different media assumptions (see [4]). For example, assuming that the media is isotropic with space homogeneous conductivity tensor and is an infinite region, the following formula can be easily derived:
(3)
where (Ji, (Je are the intracellular and extracellular conductivity coefficients, respectively and V is the transmembrane potential computed via bidomain or monodomain equations. From this formula, the body surface ECG is the summation of weighted difference at different space positions of heart. Moreover this formula links the microscale to macroscale. A typical ECG pattern can be reproduced by this formula as shown in Figure 4. A full description of the body surface ECG is obtained by solving the
162 EGG Pattern (normal v. s. 0.52 factor IK1 reduction)
0.8 ';' 0.6
5 0
<..)
w 0.4 0.2 0
~-~
-0.2 0
Fig. 4.
100
200
300
400
500 600 time (ms)
700
800
900
1000
EeG pattern (no p wave here due to the lack of atrium part)
following complete model ([33, 34]): X * CmaN -\7. ((Di
= -\7. (De \7u e ) -
+ De)\7ue ) =
X * lion (V, Y, 8)
\7. (Di\7V) - l~pp,
X
+ l~pp, E
X
E
R
R
atY = gl(V, Y), X E R at8 = g2(V, Y, 8, B),
X
E
R
atB = g3(8, B), X E R Di\7(V + u e ) . n T = 0, U e = u o , X E oR
X
EaR
(De \7ue ) . n T
= (Do \7u o) . nT, \7. (Do\7u o) = 0, X E T (Do \7u o) . n T = 0, X EaT.
(4)
X E
oR
In short, this complete model is composed of two part: heart region and torso region. In the heart region, the governing equation is the bidomain model and in the torso region, the governing equation is the poisson equation assuming no charge source outside heart region. On the heart-torso boundary, no intracellular current flux condition and continuity of extracel-
163
lular potential and torso potential are imposed. On the torso-air boundary, no current flux condition is imposed. To date, on account of its complexity, only one group (G. Lines et al. [33]) has achieved this complete solution for realistic heart and torso geometries. They showed a single simulation of a transmembrane potential distribution in the heart along with a simultaneously-computed body surface distribution ([34]). 5. Applications in hear arrhythmia Hear arrhythmia is a fatal disease and sudden cardiac death (SCD) account for >300000 deaths per year in the USA. Therefore the probe about its origin and prevention are valuable. Due to its cheap cost, computer simulation is a good tool to explore arrhythmia. Among other things, ventricular tachycardia (VT) and ventricular fibrillation (VF) are two most dangerous abnormal electrical events. The whole heart scale ECG patterns of VT, VF can be linked to the tissue scale electrical activities as shown in Figure 5. From the simulation results in Figure 5, it can be easily seen that VT is induced by the reentry waves or spiral waves. Since the frequency of reentry waves is higher than the normal sinus rhythm, the hearts of patients with VT beat faster. Also, VF is induced by the broken wavelets which may be due to the breakup of reentry waves. Therefore, many researchers have payed much attention on the origins of reentry waves and their breakup ([16, 66, 67]). From computation formula (3), we can also discuss the cellular basis of various pathological electrophysiology of heart. Various pathological conditions can also induce arrhythmia, e.g. ischemia, hypertension, hypertrophy, infarction ... Also, computerr simulation is a good tool to discuss the effect of these pathological factors on heart functions. For example, in [73] they used the conduction restitution and effective refractory period (ERP) restitution properties to explore the arrhythmogenic ingredients of ischemia. Recently the electrophysiological role played by the pulmonary veins sleeves in the atrial fibrillation is a hot issue ([63, 64]). In [35], the computer simulation approach also was used to analyze the corresponding phenomena. 6. Conclusion Heart modeling is a task of high challenge. Our goal is to build up a patient specific effective simulation system. The input of the system is the individual patient image data from CT, MRI, ultrasound or others and the
164
Fig. 5. Heart arrhythmia: events relation between tissue scale and whole heart scale (adapted from http://arrhythmia.hofstra.edu./).
output is the computer simulations results. With such system, it will be easier to make correct clinical diagnosis and prognosis. However, we are far away from this perfection. Up to now, it still consumes too much time to simulate heart dynamics and so is ineffective. Furthermore, the interplay between electrophysiology, muscle mechanics, and hemodynamics should be considered simultaneously and its progress is far insufficient now. For the coupling of electrophysiology and mechanics, there are two folds. On one hand, the electrical excitation will induce mechanical contractionIn (Ee coupling, see [2]). On the other hand, mechanical stretch will induce some ionic currents which then have feedback on the electrophysiology (MEF, [29-31]) and the heart geometry will change in time by the constitutive strain-stress relation which then promotes the difficulty of electrophysiological simulation. In this article we just mentioned some important topics in the electrophysiology aspect. About the muscle mechanics, reference [47J by J.J.Rice is a good review article. There are still many problems remained to overcome in the heart modeling. References 1. Aliev RR, Panfilov AV. A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals 1996; 7:293-301. 2. Bers DM. Excitation-Contraction Coupling and Cardiac Contractile Force, 2nd edition. Kluwer Academic Publishers, 2001.
165 3. Beeler GW, Reuter H. Reconstruction of the action potential of ventricular myocardial fibres. J Physiol. 1977 Jun;268(1):177-21O. 4. Bin He. Modeling and imaging of bioelectrical activity: principle and applications. New York: Kluwer Academic/Plenum Publishers, 2004. 5. Bruaset AM, Tveito A, editors. Numerical solution of partial differential equations on parallel computers. Springer, 2006. 6. Buist M, Sands G, Hunter P, Pullan A. A deformable finite element derived finite difference method for cardiac activation problems. Ann Biomed Eng. 2003 May;31 (5):577-88. 7. Chernyak YB, Feldman AB, Cohen RJ. Correspondence between discrete and continuous models of excitable media: trigger waves. Phys Rev E 1997; 55:3215-33. 8. Colli Franzone P, Pavarino LF, Taccardi B. Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models. Math Biosci. 2005 Sep;197(1):35--66. 9. Courtemanche M, Ramirez RJ, Nattel S. Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am J Physiol. 1998 Jul;275(1 Pt 2):H301-21. 10. DiFrancesco D, Noble D. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond B Bioi Sci. 1985 Jan 10;307(1133):353-98. 11. Fast VG, Efimov IG. Stability of vortex rotation in an excitable cellular medium. Physica D 1991; 49:75-81. 12. Feldman AB, Chernyak YB, Cohen RJ. Cellular automata model of cardiac excitation waves. Herzschr Elektrophys 1999; 10:92-104. 13. Feldman AB, Chernyak YB, Cohen RJ. Wave-front propagation in a discrete model of excitable media. Physical Review E 1998; 57:7025-7040. 14. Fenton FH, Cherry EM, Hastings HM, Evans SJ. Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 2002; 12:852-892. 15. Fernandez JW, Mithraratne P, Thrupp SF, Tawhai MH, Hunter PJ. Anatomically based geometric modelling of the musculo-skeletal system and other organs. Biomech Model Mechanobiol. 2004 Mar;2(3):139--55. 16. Garfinkel A, Kim H, Voroshilovsky 0, et al. Preventing ventricular fibrillation by flattening cardiac restitution. Proc Natl Acad Sci USA 2000; 97:6061-6. 17. Gerhardt M, Schuster H, Tyson JJ. A cellular automaton model of excitable media including curvature and dispersion. Science 1990; 247: 1563-1566. 18. Greenberg JM, Hastings SP. Spatial patterns for discrete models of diffusion in excitable media. SIAM Journal on Applied Mathematics 1978; 34:515-523. 19. Greenstein JL, Winslow RL. An integrative model of the cardiac ventricular myocyte incorporating local control of Ca2+ release. Biophys J. 2002 Dec;83(6):2918-45. 20. Harrild DM, Penland RC, Henriquez CS. A flexible method for simulating cardiac conduction in three-dimensional complex geometries. J Electrocardiol. 2000 JUl;33(3):241-51.
166 21. Hille B. Ion channels of excitable membranes, 3rd edition. Sunderland: Sinauer Associates, Inc., 2001. 22. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 1952; 117:500-544. 23. Huguenard JR, Prince DA. A novel T-type current underlies prolonged Ca(2+ )-dependent burst firing in GABAergic neurons of rat thalamic reticular nucleus. J Neurosci. 1992;12(10):3804-17. 24. Huguenard JR, McCormick DA. Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons. J Neurophysiol. 1992 Oct;68(4):1373-83. 25. Iyer V, Mazhari R, Winslow RL. A computational model of the human leftventricular epicardial myocyte. Biophys J. 2004 Sep;87(3):1507-25. 26. Jafri MS, Rice JJ, Winslow RL. Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys J. 1998 Mar;74(3): 1149-68. 27. Karma A. Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys Rev Lett. 1993 Aug 16;71(7):1103-1106. 28. Keener J, Sneyd J. Mathematical Physiology. NewYork:Spinger-Verlag, 1998. 29. Kohl P, Ravens U. Cardiac mechano-electric feedback: past, present, and prospect. Prog Biophys Mol BioI. 2003 May-Jul;82(1-3):3-9. 30. Kohl P and Sachs F. Mechanoelectric feedback in cardiac cells. Phil. Trans. R. Soc. Lond. A 2001;359:1173-1185. 31. Lab MJ. Monophasic action potentials and the detection and significance of mechanoelectric feedback in vivo. Prog Cardiovasc Dis. 1991 JulAug;34(1):29-35. 32. LeG rice IJ, Smaill BH, Chai LZ, Edgar SG, Gavin JB, Hunter PJ. Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Am J Physiol. 1995 Aug;269(2 Pt 2):H571-82. 33. Lines GT, Grottum P, Tveito A. Modeling the electrical activity of the heart: A bidomain model of the ventricles embedded in a torso. Comput. Visual. Sci. 2003; 5:195-213. 34. Lines G, Buist ML, Grottum P, Pullan AJ et al. Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Visual. Sci. 2003; 5:215-239. 35. Lo CP, Horng TL, Luk HN et al. A computational model of the canine pulmonary veins sleeves. Journal of medical and biological engineering 2006; 26(3):125-130. 36. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 1994; 74:1071-96. 37. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation. Circ Res 1994; 74:1097-133. 38. Markus M. Dynamics of a cellular automaton with randomly distributed
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170
SYMMETRY PROPERTIES OF POSITIVE SOLUTIONS OF PARABOLIC EQUATIONS: A SURVEY P. Polacik
School of Mathematics, University of Minnesota Minneapolis, MN 55455 This survey is concerned with positive solutions of nonlinear parabolic equations. Assuming that the underlying domain and the equation have certain reflectional symmetries, the presented results show how positive solutions reflect the symmetries. Depending on the class of solutions considered, the symmetries for all times or asymptotic symmetries are established. Several classes of problems, including fully nonlinear equations on bounded domains, quasilinear equations on JRN, asymptotically symmetric equations, and cooperative parabolic systems, are examined from this point of view. Applications of the symmetry results in the study of asymptotic temporal behavior of solutions are also shown.
Keywords: Parabolic equations and systems, positive solutions, symmetry, moving hyperplanes, asymptotic behavior
1. Introduction: basic problems, results and some history A conventional wisdom says that "parabolic flows reduce complexity." Although it should not be taken too seriously and universally, there are good examples where it manifests itself. Asymptotic symmetry of solutions, which is one of the main topics of this survey, is such an example. It shows a tendency of positive solutions of certain parabolic equations to "improve their symmetry" as time increases, becoming "symmetric in the limit" as t -+
00.
Historically, first studies of symmetry properties of positive solutions of parabolic equations were carried out after similar properties for elliptic equations had been long understood. These studies brought about interesting qualitative results for parabolic equations, but at the same time they opened new perspectives for looking at the earlier results for elliptic equations. Viewing solutions of elliptic equations as equilibria (steady states) of the corresponding parabolic equations, one naturally tries to understand
171
how their symmetry fits in the broader picture of the parabolic semiflow. For example, examining heteroclinic orbits between symmetric equilibria, one naturally asks if they are symmetric as well. Generalizing, one is subsequently lead to the problem of symmetry of entire solutions, that is, solutions defined for all times, positive and negative. Another symmetry problem is concerned with general solutions of parabolic equations, which are typically defined for positive times only. If the parabolic semiflow admits a Lyapunov functional, which forces bounded solutions to converge to steady states, the symmetry of the latter translates to the asymptotic symmetry of the solutions of the parabolic equation. This immediately raises the question whether the asymptotic symmetry can be established regardless of the presence of any Lyapunov functional, even if the solution does not converge to a steady state. This question is even more interesting for time-dependent parabolic problems, whose solutions typically do not converge to equilibria and their temporal behavior can be very complicated. In this case, the asymptotic symmetrization in space is to be studied independently of the temporal structure. Once it is understood, however, it often proves very useful for studying the temporal behavior of the solutions. The previous paragraph indicates the sort of problems this survey is devoted to. Considering parabolic equations with certain symmetry properties, we want to understand how their solutions reflect the symmetry. The key issues to be discussed are the asymptotic symmetry properties for the Cauchy problem, symmetry of the entire solutions (and related to this, symmetry of unstable spaces of entire solutions), and applications of these results. We also mention key ideas of the proofs and discuss differences of their use for different type of problems. To give a more specific overview of the results to be presented in this paper and to put them in context with similar results on elliptic equations, we first consider the following semilinear reaction-diffusion equation Ut
= t>.u + f(t, u),
xE
n, t
E J.
(1.1)
Here n is a domain in ]R.N and f : ]R.2 ...... ]R. is a continuous function, which is Lipschitz continuous with respect to u. We take either J = (0,00) and consider a suitable initial-value problem, or J = (-00,00) in case we want to consider entire solutions. Although much simpler an equation than fully nonlinear ones examined is the forthcoming sections, for the purposes of the introduction (1.1) is sufficiently representative and it allows us to discuss some key issues in a more rudimentary way. The simplicity of (1.1) consists mainly in the fact
172
that, without any additional assumptions, the equation is invariant under any Euclidean symmetries the domain may have. This is true regardless of the temporal dependence of f on time, which is not restricted by any assumption like periodicity or almost periodicity. We assume that either D = ]RN, or D is a bounded domain in ]RN which is symmetric with respect to the hyperplane
and convex in Xl (which means that with any two points (Xl, X'), (Xl, X'), differing only at the first component, D contains the line segment connecting them). The specific choice of the direction el for the reflectional symmetry is arbitrary, domains symmetric in other directions can be considered equally well. In case D is bounded, we complement the equation with Dirichlet boundary condition
u(X, t) = 0,
x
oD, t E J.
E
(1.2)
If f is independent of t, steady states of (1.1), (1.2) are solutions of the elliptic equation ~u+f(u)=O,
xED,
(1.3)
complemented (in case D is bounded) with the Dirichlet condition
u(X) =0,
xEoD.
(1.4)
Symmetry theorems for (1.1) have somewhat different flavors and hypotheses for bounded domains D and for D = ]RN, so we distinguish these case separately. 1.1. Equations on bounded domains
Reflectional symmetry of positive solutions of (1.3), (1.4) was first established by Gidas, Ni and Nirenberg. 37 They proved that if D is as above (convex and symmetric in Xl) and smooth (of class C 2 ), then each positive solution u of (1.3), (1.4) has the following symmetry and monotonicity properties: U(-Xt,X2, ... ,XN) =U(Xl,X2, ... ,XN) U X1 (Xl,X2, ... ,XN)
<
° (x
E
D,
Xl
(x ED),
> 0).
(1.5)
The method of moving hyperplanes, which is the basic geometric technique in their paper, was introduced earlier by Alexandrov 2 and further
173
developed by Serrin70 (Ref. 70 also contains a related result on radial symmetry). Generalizations and extensions of the symmetry result have been made by many authors. In particular, Li50 extended it to fully nonlinear equations on smooth domains. Later Berestycki and Nirenberg l l found a way of dealing with fully nonlinear equations on nonsmooth domains employing the elliptic maximum principle for domains with small measure (see also Dancer's contribution in Ref. 29, where semilinear equations on nonsmooth domains are treated using a different method). There are many other related results, including further developments regarding symmetry of elliptic overdetermined problems, as considered in the original paper of Serrin,70 see for example Refs. 1,73. Additional references and more detailed overviews can be found in the surveys Refs. 7,46,58. Let us also mention a more recent work by Da Lio and Sirakov, Ref. 27, where the symmetry results are extended to viscosity solutions of general elliptic equations. The proofs of the above results are based on the method of moving hyperplanes and various forms of the maximum principle. Below we shall indicate how these techniques are typically used. A different approach employing a continuous Steiner symmetrization was used in Ref. 12 (see also the survey Ref. 46). It applies to positive solutions of (1.3), (1.4) and, as it relies on the variational structure of the problem and not so much the maximum principle, it allows for extensions of the results of Ref. 37 in different directions. We remark that it is not always possible to generalize the above symmetry result to solutions which are merely nonnegative, rather than strictly positive, (counterexamples are easily constructed if N = 1). However, if N > 1 and some regularity assumptions are made on the domain, it can be proved that nonnegative solutions are necessarily strictly positive, hence symmetric (see Ref. 19,33 for results of this sort). The issue whether some strict positivity assumption is needed or not will arise again in our discussion of parabolic equations below. Also the convexity of n in Xl is an important assumption without which the result is not valid in general (however, see Ref. 47 for a symmetry result involving some nonconvex domains). If n is a ball, say n = B(O, TO) (0 is the center, TO the radius), then the reflectional symmetry theorem can be applied in any direction which leads to the following radial symmetry result. Any positive solution u(x) of (1.3), (1.4) is radially symmetric (it only depends on T = Ix!) and radially decreasing (ur(x) < 0 for T E (O,TO)). There are numerous application of the above symmetry results in further studies of positive solutions of (1.3), (1.4). For example, the radial symme-
174
try property implies that positive solutions can be viewed as solutions of the ordinary differential equation (ODE) N-1
U rr
+ - -r Ur + feu)
= 0,
r E (0, ro),
(1.6)
and that ur(O) = 0. Thus one immediately gains ODE tools, like the shooting method, for the study of positive solutions. Problems on multiplicities and/or bifurcations of positive solutions become then a lot more elementary. The reflectional symmetry results do not lead to such dramatic simplifications of the problem, but they are still very useful, especially if there are several directions in which the domain is symmetric. For parabolic problems, such as (1.1), (1.2), first symmetry results of similar nature started to emerge much later. After a prelude30 devoted to time-periodic solutions, symmetry of general positive solutions of parabolic equations on bounded domains was considered in Refs. 4,5,44 and later in Refs. 6,63. With 0 as above (convex and symmetric in Xl) and with suitable symmetry assumptions on the nonlinearity, symmetries of two classes of solutions were examined in these papers. Closer in spirit to the results for elliptic equations are symmetry theorems concerning entire solutions. A typical theorem in this category states that if U is a bounded positive solution of (1.1), (1.2) with J = lR satisfying inf u(x, t) >
tElR
° (x
E
0, t E J),
(1.7)
then u has the symmetry and monotonicity properties (1.5) for each t E R
u( -Xl, X', t) U X1
= U(Xl' x', t) (x =
(x, t) <
° (x E 0,
Xl
(Xl,
> 0, t
x') EO, t E lR),
E lR).
(1.8)
This result follows from more general theorems of Refs. 4,6, although to be precise we would need to include additional compactness assumptions on U (in the context of the semilinear problem (1.1), (1.2), the boundedness ofu alone is sufficient if, for example, 0 has Lipschitz boundary and t - f(O, t) is a bounded function). In a different type of symmetry results, nonnegative solutions of the Cauchy-Dirichlet problem for (1.1) are considered. These of course cannot be symmetric, unless they start from a symmetric initial function. However, it can be shown that they "achieve" the symmetry in the limit as t - 00. To formulate this more precisely assume that U is a bounded positive solution of (1.1), (1.2) with J = (0,00) such that for some sequence tn - 00
liminfu(x,tn) > n-+oo
° (x EO).
(1.9)
175
Then
U
is asymptotically symmetric in the sense that lim (u( -Xl, X', t) - U(XI, X', t)) = 0
t ..... oo
limsupuxJx,t) t ..... oo
~ 0
(X E !l), (X E!l,
Xl>
0).
(1.10)
If {u(·, t) : t 2: I} is relatively compact in C(D), then the asymptotic symmetry of U can be expressed in terms of its limit profiles, that is, elements of its w-limit set,
w(U):= {¢: ¢ = limu(,tn) for some tn
~
oo},
where the limit is in C(D) (with the supremum norm). It can be easily shown that condition (1.9) is equivalent to the requirement that there exist at least one element of w(u) which is strictly positive in!l and (1.10) translates to all elements of w(u) being symmetric (even) in Xl and monotone nonincreasing in Xl > O. This result is proved in a more general setting in Ref. 63. Condition (1.9) is a relatively minor strict positivity condition (note that it is not assumed to be valid for all sequences tn ~ 00), which cannot be omitted in general (a counterexample can be found in Ref. 63). It is not needed, however, if the domain is sufficiently regular. 44 There are connections between the two types of symmetry results, the asymptotic symmetry for the Cauchy-Dirichlet problem and the symmetry for entire solutions. Oftentimes, ifthe nonlinearity is sufficiently regular, the limit profiles of a solution of the Cauchy-Dirichlet problem can be shown to be given by entire solutions of suitable limit parabolic problems. Thus if the limit profiles are all positive (i.e., (1.9) holds for any sequence tn ~ 00), the symmetry of entire solutions for the limit problems can be used to establish the asymptotic symmetry of positive solutions of the original Cauchy-Dirichlet problem. This is how the asymptotic symmetry is proved in Refs. 4,6. A different approach to asymptotic symmetry is used in Ref. 63. It is based on direct estimates, not relying on any limit equation, thus the regularity and positivity requirements on the nonlinearity and the solutions are significantly relaxed compared to the earlier results. A yet different approach was used in the original paper Ref. 44. While it requires more regularity of the nonlinearity and the domain, it does not assume any strict positivity condition. Also it has an interesting feature in that it shows that the symmetry of a positive solution u improves with time in the sense that a quantity which can be thought of as a measure of symmetry increases strictly along any positive solution which is not symmetric from the start. See Section 3 for precise formulations of the above results in the context of fully nonlinear parabolic equations. Results on entire solutions given
176
there also include a statement on the symmetry of unstable spaces of positive solutions. In the special case when the positive solution is a steady state of a time-autonomous equation, the statement says that the eigenfunctions of the linearization around the steady state corresponding to negative eigenvalues are all symmetric. When considering the asymptotic symmetry of solutions, several natural questions come to mind. For example, can the asymptotic symmetry be proved if the equations itself is not symmetric, but rather is merely asymptotically symmetric as t -+ oo? One could for example think of equations (1.1) with an extra term added, say Ut
= ~u + f(t,u) + g(t,u),
x E n,t > 0,
where get, u) -+ 0 as t -+ 00 for each u. Then one can also consider relaxing other conditions, like the assumption of positivity of the solutions, and only require them to be satisfied asymptotically. Sometimes such problems can be addressed in a relatively simple manner. Indeed, as in the discussion above, if the limit profiles of a solution considered can be shown to be given by positive entire solutions of a limit equation, then, the limit equations being symmetric by assumption, one can apply to these entire solution the symmetry results discussed above. This gives the asymptotic symmetry of the original solution. However, in a general setting such a simple argument may not be applicable, a simple possible reason being that the original solution does not have a strictly positive inferior limit as t -+ 00 at each point x E n. In that case, not only is the treatment of asymptotically symmetric problems more complicated, the result may not be true in the form one could expect. Asymptotically symmetric problems are considered in the recent work Ref. 35. We include statements of the main theorems and some discussion in Section 3.3.
1.2. Equations on
]RN
Let us now take n = ]RN. In their second symmetry paper Ref. 38, a sequel to Ref. 37, Gidas, Ni and Nirenberg considered elliptic equations on ]RN including the following one (1.11) They assumed that f(O) = 0 and made other hypotheses on the behavior of feu) near u = o. They proved that each positive solution u(x) of (1.11) which decays to 0 as Ixl -+ 00 at a suitable rate has to be radially symmetric around some ~ E ]RN and radially decreasing away from ~. Later it was
177
proved by Li and Ni 52 that a mere decay (with no specific rate) is sufficient for the symmetry of u if f(O) = and l' is nonpositive near zero (under the stronger condition 1'(0) < 0, this result was also proved in Ref. 51). In both Ref. 51 and Ref. 52, general fully nonlinear equations satisfying suitable symmetry assumptions are treated. Many other extensions of the symmetry results are available. For example, one can consider some degenerate equations 71 on ]RN or different types of unbounded domains. 8 - 10 ,69 Again we refer the reader to the surveys Refs. 7,58 for more details and references. Contrary to elliptic equations, symmetry results for parabolic equations on ]RN did not appear so soon after the first results on bounded domains. The fact that the possible center or hyperplane of symmetry in ]RN is not fixed a priori (unlike on bounded domains) adds an interesting flavor to the symmetry problem and is the cause of major difficulties. Already in the simple autonomous case, the problem is by no means trivial. Consider for example the Cauchy problem
°
Ut
=
tlu + f(u),
u=uo,
x E ]RN, x
t > 0,
E ]RN, t = 0,
(1.12)
where f is of class C1, f(O) = 0, and Uo is a positive continuous function on]RN decaying to at Ixl = 00. Assume the solution u of (1.12) is global, bounded, and localized in the sense that
°
supu(x, t) t~O
-+
°as Ixl
-+ 00.
(1.13)
It is not clear whether u is asymptotically radially symmetric around some center, even if it is known that its w-limit set w(u) consists of steady states, each of them being radially symmetric about some center. It is not obvious whether all the functions in w( u) share the same center of symmetry and, in fact, that is not true in general. A counterexample can be found in Ref. 68 where equations (1.12) with N ~ 11, f(u) = uP, and p sufficiently large are considered. The proof of the existence of a solution with no asymptotic center of symmetry, as given there, depends on the fact that the steady states, in particular the trivial steady state, are stable in some weighted norms but are unstable in Loo(]RN) (see Refs. 40,41,67). If, on the other hand, one makes the assumption 1'(0) < 0, which in particular implies that u == is asymptotically stable in Loo(]RN), then bounded solutions satisfying (1.13) do symmetrize as t -+ 00: they actually converge to a symmetric steady state. This convergence result is proved in Ref. 15, under slightly stronger hypotheses (exponential decay of the solution at spatial
°
178
infinity); for more specific nonlinearities proofs can also be found in Refs. 26, 34. The proofs of these convergence theorems depend heavily on energy estimates and are thus closely tied to the autonomous equations. The symmetry problem for nonautonomous parabolic equations on n = ]RN, such as (1.1), was addressed in Refs. 61,62. The asymptotic symmetry for solutions of the Cauchy problems as well as symmetry for all times for entire solutions is established in these papers, see Section 4 for the statements. It is worthwhile to mention that these result are available for quasilinear equations only, not for fully nonlinear as in the case of bounded domains. The technical reasons for this will be briefly explained in Section 4.
1.3. Cooperative systems We shall now discuss extensions of the symmetry results to a class of parabolic systems. A model problem is the following cooperative system of reaction-diffusion equations Ut
= D(t)~u+
f(t,u),
(x,t) En
X
(0,00).
(1.14)
Here D(t) = diag(d 1 (t), ... ,dn(t» is a diagonal matrix whose diagonal entries are continuous functions bounded above and below by positive constants, and f = (II, ... , fn) : [0,(0) X R.n ---4 ]Rn is a continuous function which is Lipschitz continuous in u E ]Rn and which satisfies the cooperativity condition ali(t, u)/aUj :::: whenever i =I- j and the derivative exists (which is almost everywhere by the Lipschitz continuity). We couple (1.14) with the Dirichlet boundary conditions
°
Ui(X,
t)
= 0,
(x, t)
E
an X
(0, (0), i = 1, ... , n.
(1.15)
In case the diffusion coefficients D and the nonlinearity are timeindependent and only steady state solutions are considered, we are lead to the elliptic system D~u+
feu)
= 0,
x E n,
(1.16)
with Dirichlet boundary conditions. SymIT'etry properties of positive solutions for such elliptic cooperative systems were established by'Iroy,75 then by Shaker 72 (see also Ref. 24) who considered equations on smooth bounded domains. In Ref. 31, de Figueiredo removed the smoothness assumption on the domain in a similar way as Berestycki and Nirenberg l l did for the scalar equation. For cooperative systems on the whole space, a general symmetry result was proved by Busca
179
and Sirakov 16 (an earlier more restrictive result can be found in Ref. 32). The cooperativity hypotheses which is assumed in all these references cannot be removed. Without it, neither is the maximum principle applicable nor do the symmetry result hold in general (see Ref. 17 and Ref. 72 for counterexamples) . Parabolic cooperative systems, such as (1.14), (1.15), were considered in Ref. 36, where the asymptotic symmetry of positive solutions is proved (see Section 5 for the results). A new difficulty that arises when dealing with parabolic systems, as opposed to scalar parabolic equations or elliptic systems, is that different components of the positive solution may be very small at different times. This situation has to be handled carefully using Harnack type estimates which were developed in Ref. 36 for this purpose. Similar symmetry results for parabolic systems on ]RN can be proved using ideas form Ref. 36 and Ref. 61, but they are not documented in literature.
1.4. Applications When it comes to applications of the symmetry results in further qualitative studies of parabolic equations, the matters are more complicated than in the case of elliptic equations. Even when dealing with radially symmetric solutions of (1.1), the analogue of (1.6) is Ut
N -1 r
= U rr + - - U r + f(t, u) = 0,
(1.17)
which is still a PDE. Even worse, studying positive solutions of the Cauchy problem, we only have the asymptotic symmetry results, hence (1.17) can only be valid asymptotically, if one can make a sense of that. Nonetheless, the symmetry theorems have proved very useful for further studies of positive solutions of parabolic problems. For example, they have been used in the proofs of convergence results for some autonomous and time-periodic equations. We will sketch the proof of such a convergence theorem in Section 7. In that section we also discuss some open problems related to symmetry properties of positive solutions and indicate possible directions of further research. We would like to emphasize that we have devoted this survey exclusively to symmetry properties related to the positivity of solutions. There are many other results in literature where symmetry is shown to be a consequence of other properties of solutions, like stability (see, for example, Refs. 57,59 and references therein) or being a minimizer for some variational
180
problems (see Ref. 55 and reference therein). Different types of parabolic symmetry results can also be found in Refs. 25,45,56,69.
2. General notation The following general notation is used throughout the paper. For Xo E ]RN and r > 0, B(xo, r) stands for the ball centered at Xo with radius r. For a set 0 C ]RN and functions v and w on 0, the inequalities v ~ and w > are always understood in the pointwise sense: v(x) ~ 0, w(x) > (x EO). For a function z, z+, z- stand for the positive and negative parts of z, respectively:
° °
z+(x) = (lz(x)1
+ z(x»/2
z-(x) = (lz(x)l- z(x»/2
°
~ 0, ~
0.
If Do, D are subsets of]Rm with Do bounded, the notation Do cc D means Do cD; diam(D) stands for the diameter of D; and IDI for the (Lebesgue) measure of D (if D is measurable). In each section of the paper, 0 is a fixed domain in ]R N and we denote £:= SUp{XI: (XI'X') EO for some x' E ]RN-I}::; 00,
0>. H>.
{x EO: Xl > >..}, := {x E]RN : Xl = >..}, :=
(2.1)
r>.:= H>. nn. By P>. we denote the reflection in the hyperplane H>.. Note that if 0 is convex in Xl and symmetric in the hyperplane Ho, then P>.(O>.) C 0 for each>" E [0,£). For a function z(x) = Z(XI'X') defined on 0, let z>. and V>.z be defined by
z>'(X) = z(P>.x) = z(2).. - XI,X'), V>.z(x) = z>'(x) - z(x)
(x EO>.).
(2.2)
3. Fully nonlinear equations on bounded domains
In this sE-ction we consider fully nonlinear parabolic problems of the form Ut
=
F(t,x, u, Du, D 2 u),
u = 0,
x E 0, t E J, x E
a~,
t E J.
(3.1) (3.2)
Here 0 C ]RN is a bounded domain and J is either (0,00) or (-00, T) for some T ::; 00. Taking J = (0,00) we have the Cauchy-Dirichlet problem in mind, although usually we do not write down the initial condition explicitly
181
(it does not playa role in our analysis). Included in the case J = (-00, T) are entire solutions (T = 00), but the results we state apply also to T < 00. We make the following assumptions (D1)
nc
]RN is a bounded domain which is convex in Xl and symmetric about the hyperplane Ho = {x = (Xl, x') E]RN : Xl = O}: {(-'-XI,X'): (XI,X') En}
=
n.
(D2) For each A > 0, the set
n,\ := {x En: Xl > A} has only finitely many connected components. Via a canonical isomorphism, we identify the space of N x N-matrices 2 with ]RN . The nonlinearity F: (t,x,u,p,q) J-+ F(t,x,u,p,q) is defined on J x x B, where B is an open convex set in]R x]RN X ]RN2 which is invariant under the transformation Q defined by
n
Q(u,p, q) = (u, -P!'P2,.·· ,PN, ij), _ _ {-qi j qij
%-
(3.3)
if exactly one of i, j equals 1,
.
otherwIse.
We assume that F satisfies the following conditions:
n
(F1) (Regularity) F is continuous on J x x B and Lipschitz in (u,p, q), uniformly with respect to (x, t): there is (3 > 0 such that sup
IF(t,x,u,p,q) - F(t,x,u,p,q)1
~
(31(u,p,q) - (u,p,q)1
xEO,t2':O
«u,p, q), (u,p, q) Moreover, F is differentiable with respect to q on J x (F2) (Ellipticity) There is a constant c¥o > 0 such that F%(t,x,u,P,q)~i~j ~ c¥ol~12
«t,x,u,p,q)
E J
E
B).
(3.4)
n x B.
x n x B, ~
E
]RN).
(3.5) Here and below we use the summation convention (summation over repeated indices). For example, in the above formula the left hand side represents the sum over i,j = 1, ... , N. (F3) (Symmetry and Monotonicity) For any (t, u,p, q) E J x B and any (Xl, X'), (Xl, x') En with Xl > Xl ~ 0 one has
F(t,±xI,X',Q(u,p,q)) = F(t,x!,x',u,p,q) ~ F(t,XI,X',u,p,q).
182
We consider global classical solutions u of (3.1), (3.2). By this we mean functions u E C 2 ,1(D x J) n C(O x J) such that
(u(x, t), Du(x, t), D 2 u(x, t))
E
B
(x
E
D, t E J)
and (3.1), (3.2) are satisfied everywhere. We assume the following conditions on u. (U1) lIu(·, t)IIL=(O) is bounded uniformly in t E J. (U2) The family of functions u(·,· + s + 1), s E J, is equicontinuous on D x [0,1]: lim
sup
lu(x,t+s+1)-u(x,t+s+1)1=0.
h ..... O x,xEn, t,tE[O,l],
[x-x/,[t-li
Let us make some comments on our hypotheses and overall set up of the problem. Hypothesis (D2), which rules our some very irregular domains D, is assumed for technical reasons and we do not know if it can be removed from the hypotheses or not (it is not needed in some theorems, as explicitly indicated). The differentiability requirement on F with respect to q can be relaxed somewhat (see Remark 2.1 in Ref. 63). Also it should be noted that although we assume the global Lipschitz continuity of F (and similarly for the ellipticity), it is clear that we only need this assumption on the range of (u, Du, D 2 u) for a solution u considered. If the range is bounded, then local Lipschitz continuity, uniform in x and t, is a sufficient assumption. However, we would like to emphasize that only the boundedness of u, and not of its derivatives, is required in (U1). While it is often very easy to obtain a bound on a solution u of a Cauchy problem using a comparison argument, bounds on its derivative are usually not so easy to find, in particular if the equation is fully nonlinear. In this connection, we would also like to stress that in this paper we are not concerned with problems of global existence of solutions. Rather, given a global solution, we want to understand if there are consequences of its positivity on its symmetry properties. It is a different question whether a local solution u of an initial value problem which is only assumed bounded in LOO(D) is actually global (after the maximal extension of the existence time interval). This is always true in semilinear equations under suitable growth conditions (see Ref. 3 for example), but not in fully nonlinear equations. Still, it is of interest to derive a priori information on the behavior of a bounded solution u which is assumed global. The equicontinuity condition (U2) presents no extra restriction if (0,0,0) E B, F(x, t, 0, 0, 0) is bounded, and D satisfies minor regularity
183
conditions (Lipschitz continuity is sufficient, a weaker sufficient condition is Condition (A) of Ref. 49). As shown in Proposition 2.7 of Ref. 63, (U2) then follows from (U1) and parabolic boundary Holder estimates. Without any regularity of n, (U2) can be shown to hold if (U1) holds together with the following stronger form of the boundary condition: (U2)' u(x, t)
---->
0 as dist(x, an)
---->
0, uniformly with respect to t E J.
We focus on reflectional symmetries of solutions, but, as noted in the introduction, if the domain and the equation are invariant under all rotations ofll~N (or only rotations of a subspace ofll~N), then reflectional symmetries in all admissible variables can be used to establish a rotational symmetry of positive solutions. We formulate, as an illustration, one result on radial symmetry in the next subsection, but refrain from giving such standard corollaries in the other subsections.
3.1. Solutions on (0,00) In this subsection we take J = (0,00). Hypotheses (U1), (U2) imply in particular that {u(·, t) : t > I} is relatively compact in C(!1). We introduce the w-limit set of u in this space:
w(u)
= {z
E
C(!1) : ilU(·,tk) -
ziiL=(O) ---->
0 for some tk
---->
oo}.
Observe that {u(·,t): t > I} is also compact in C o(!1), the closed subspace of C(!1) consisting of all continuous functions on!1 vanishing on an. Hence w(u) C C o(!1). Also, distc(o)(u(.,t),w(u))
---->
0 as t
---->
00.
We can now state the first asymptotic symmetry result.
Theorem 3.1. Let (D1), (D2), (F1)-(F3) hold and let u be a nonnegative solution of (3.1), (3.2) satisfying (U1), (U2). Assume that there is 4> E w(u) such that 4> > 0 on n. Then u is asymptotically symmetric and monotone in Xl. More specifically, for each z E w( u) one has (3.6)
and either z == 0 or else z is strictly decreasing in Xl on no. The latter holds in the form ZXl < 0 on no, provided ZXl E C(no). This is the same as Theorem 2.2 of Ref. 63. Note that without extra conditions, like boundedness of spatial derivatives of u, we cannot in general assume that elements of w(u) are differentiable. We have ZXl E C(no) for
184
each Z E w(u), provided {U X1 (., t) : t > 1} is relatively compact in C(13) for each closed ball 13 c no. We remark that the assumption that ¢ > for some ¢ E w(u) cannot be removed even if u is strictly positive. A counterexample is given in Ref. 63. It consists in finding a solution u of the semilinear problem
°
ut=b..u+f(x2,u) u
= 0,
xEn,t>O, x E
(3.7)
an, t > 0,
(3.8)
where n := (-1,1) x (-1,1) and f is a suitable Lipschitz nonlinearity, with the following properties: u is (strictly) positive in n x (0,00) and u(·, t) ____ Z in CoCO) where Z E CoCO) is a nonnegative function such that z > in no and Z(0,X2) = for each X2 E [-1,1). In particular, z is not monotone in Xl> 0. Observe that Theorem 3.1 in particular implies that if some ¢ E w(u) is positive in n then each ¢ E w( u) \ {O} is positive in n. One can give several sufficient conditions for the existence of a positive element of w(u). For example, Theorem 2.5 of Ref. 63 says that if (0,0,0) E Band
°
°
lim inf F(t, x, 0, 0, 0) 2 0,
xEn, t-+oo
°
then either w(u) = {O} or else there exists ¢ E w(u) with ¢ > in n. Also notice that if n is a ball, as in Corollary 3.3 below, then the assumption that ¢ > for some ¢ E w(u) is not needed. On more general domains, the following partial symmetry can be proved without that assumption.
°
Theorem 3.2. Let (DI), (FI)-(F3) hold and assume that nA is connected for each'\ E (0,£). Let u be a nonnegative solution of (3.1), (3.2) satisfying (UI), (U2). Then there exists ,\ ::::: such that for each Z E w(u) the following is true: z is monotone nonincreasing in Xl on n>. and
°
(3.9) A slightly more general version of this theorem is stated as Theorem 2.4 of Ref. 63. Let us now state one result, taken from Ref. 63, on radial symmetry, assuming n is a ball. Other problems in which n is rotationally symmetric and convex in some variables only can be formulated analogously. The proof of such results are rather standard: one applies the reflectional symmetry results in all admissible directions.
185
Thus let problem
n
be the unit ball centered at the origin and consider the
Ut
U
=
f(t, lxi, u, IV'ul, 6.u)
= 0,
n, t > 0, E en, t > 0,
x E
(3.10)
x
(3.11)
where f(t,r,u,'T],~) is defined on [0,00) x [0,1] satisfies the following conditions:
X
IR3. We assume that f
(F1 )rad f : [0,00) x [0, 1]
X IR3 -+ IR is continuous in all variables, Lipschitz in (u,'T],~), uniformly with respect to (x,t) and differentiable with respect to ~. (F2)rad it=. 2:: ao on [0,00) x [0,1] X IR3 for some positive constant ao. (F3)rad f is monotone nonincreasing in r.
Corollary 3.3. Let n be the unit ball, (Fl}rad-(F3}rad hold, and let u be a nonnegative solution of (3.10), (3.11) satisfying (Ul), (U2). Then u is asymptotically radially symmetric and radially nonincreasing. More specifically, each z E w( u) is a radial function and either z == or else z is strictly decreasing in r = Ix I. If z E C l (n) and z ¢. then Zr < for rE(O,l).
°
°
°
3.2. Solutions: (-00, T) and linearized problems
In this subsection we assume J = (-00, T) for some T :::; 00. The first theorem gives the symmetry of positive solutions of (3.1), (3.2) for all times t E J. Theorem 3.4. Let (Dl), (D2), (Fl}-(F3) hold and let u be a nonnegative solution of (3.1), (3.2) satisfying (Ul), (U2). Assume that there is a sequence tn -+ -00 such that liminf u(x, tn) > n--->oo
Then u is symmetric and monotone in U(-Xl,x',t)
= U(Xl,X',t)
UX1(Xl,x',t) <
°
(x E
(x
°
(3.12)
(x En).
Xl:
= (Xl,X') E n, t
n, Xl> 0, t
E J),
E J).
(3.13)
The strict positivity condition (3.12) is equivalent to the existence of a positive element of the a-limit set of u, a(u)
=
{z E C(O): Ilu(·,tk) - zllvx>(o)
-+
°for some
tk -+ -oo}.
186
The symmetry of u for all times, as given in (3.13), in particular implies that all elements of a( u) are symmetric and monotone in Xl. Theorem 3.4 is proved in Refs. 4,6 under stronger assumptions, requiring in particular that (3.12) holds for any sequence tn ~ -00 (in other words, all element of a(u) are positive). As stated, Theorem 3.4 can be proved using arguments similar to those in the proof of Theorem 3.1, as given in,63 although the general scheme of the proof has to be adjusted to solutions on (-00,0] (see Section 6 below). We next want to consider the linearization along a positive solution u of (3.1), (3.2). For that purpose we need the following additional hypothesis on F. (F4) F(t,xI,X',u,p,q) is differentiable in
XI,
u, p, q.
Assume u is a solution of (3.1), (3.2) and consider the following linearized problem
= LU(x, t)v, xED, t E (-00, T), v = 0, X E oD,t E (-oo,T),
Vt
(3.14) (3.15)
where
of LU(x, t)v := ~vxx uqij
'J
of
of
+ -;:;-VXi + ~v UPi uU
(3.16)
and the derivatives of F are evaluated at (t, x, u(x, t), Du(x, t), D 2 u(x, t)). Theorem 3.5. Let (Dl), (Fl)-(F4) hold and let u be a nonnegative solution of (3.1), (3.2) satisfying (Ul), (U2) and such that u(·, t) E C 3(D) for all t < T, the derivatives UX1t, Utxl exist and are continuous, and (3.12) holds for any sequence tn ~ -00. Let v be a solution of (3.14), (3.15) such that sup IIv(.,t)IILOO(fl) < 00.
(3.17)
t
Assume further that one of the following conditions (i)-(iii) is satisfied.
(i) IIv(·, t)IILOO(fl) ~ 0 as t ~ -00. (ii) There exist y E oD and positive constants UX1 :::; -m
(iii) There exist y
E
E,
m such that
(x E DnB(y,E),t < T).
D and positive constants
E,
FXl (t, x, u(x, t), Du(x, t), D 2 u(x, t)) :::; -m
m
such that (x E B(y, E), t
< T).
187
Then v is even in Xl: V( -Xl, X', t) = V(XI, X', t) (X E D, t < T). This is essentially a result of Refs. 5,6, although our assumptions are a little different (the arguments of Refs. 5,6 can be easily adapted). A special case of linearizations around periodic solutions was considered in Ref. 23. Let us define the center-unstable space of the solution u as the space of all bounded solutions of the linearized equation (3.14), (3.15) and the unstable space as the space of all solutions v of (3.14), (3.15) such that condition (i) holds. The previous result says that the unstable space and, under additional conditions on u or F, also the center-unstable space, consists of functions that are even in Xl. As a corollary to this theorem one can establish the symmetry of generalized eigenfunctions of the linearization at a solutions of an elliptic equation. Specifically, assuming that u is a positive steady state of a time independent problem (3.1), (3.2), consider the eigenvalue problem problem
LU(x)v + AV
= 0,
v = 0,
xED, X
E
aD,
(3.18) (3.19)
where LU(x) is as in (3.16), only now u and F are independent of t. Corollary 3.6. Let (Dl), (Fl)-(F4) hold and let F be independent of t. Let u be a positive steady state of (3.1), (3.2) such that u E C3 (D). Let v be a generalized eigenfunction of (3.18), (3.19) corresponding to an eigenvalue A with Re A ~ o. Further assume that either Re A < 0 or (Re A = 0,) v is an eigenfunction, and one of the conditions (ii), (iii) of Theorem 3.5 holds. Then v is even in Xl: v( -Xl, X') = V(XI, X') (X ED). One proves this corollary by applying Theorem 3.5 to a suitable solution of the parabolic problem (3.14), (3.15). For example, if v is an eigenfunction of (3.18), (3.19) and Re A ~ 0, then ReeAtv(x), 1m eAtv(x) are bounded solutions of (3.14), (3.15). For generalized eigenfunctions slightly more complicated formulas, involving the exponentials and polynomials of t, are used. Similar results on symmetry of real eigenfunctions of linearized elliptic problems can be derived directly by elliptic comparison arguments (see Refs. 20,28,43,54,74,76). The advantage of the approach relying on the parabolic linearized equations is that one can simultaneously treat complex eigenfunctions and generalized eigenfunctions. We remark that if none of the conditions (i)-(iii) is satisfied, the symmetry conclusion of the Theorem 3.6 may fail. Counterexamples are found in equations (1.3) with positive solutions satisfying simultaneously Dirichlet and Neumann boundary conditions.
188
3.3. Asymptotically symmetric equations In this subsection J = (0,00). We consider a nonsymmetric perturbation of (3.1), (3.2), which decays to zero as t -+ 00, making the problem asymptotically symmetric. Specifically, we consider the following problem OtU = F(t,x,u,Du,D 2 u) u(x, t)
+ Gl(x,t),
= G 2 (x, t),
(x, t) E 0 x (0,00),
(3.20)
(x, t) E 00 x (0,00),
(3.21)
where 0 and F satisfy conditions (D1), (D2), (F1)-(F3). The perturbation terms G l and G 2 are assumed to be continuous functions defined on 0 x [0,00) and 00 x [0,00), respectively, such that the following decay condition holds. (G1) G l E LN+l(O x (O,T» for each T E (0,00), and
t~ max {IIGlIILN+l(OX(t,t+l))' IIG2 (-, t)IILOO(80)} = O.
(3.22)
For some results we shall require the convergence in (3.22) to be exponential, that is, we shall assume the existence of some "I E (0,00) such that the following condition holds. (G1), G l , G 2 , are as in (G1) and there exists a positive constant C such that for each t E (0,00)
max {IIGlIILN+l(OX(t,t+l)), IIG 2 (·, t) IILOO(80)} ~ Ce-,t. Changing the constant C, the condition on G l as stated in (G1), can be equivalently formulated as IIGlIILN+l(OX(t,oo)) ::; Ce-,t
(t > 0).
We remark that the form of problem (3.20), (3.21) is general enough to cover equations with nonlinear nonsymmetric perturbation terms. For example, if Gl(x, t) in the first equation is replaced with 0 1 (x, t, u, Du, D 2 u), where Ol(X,t,U,p,q) is a function defined on x [0,00) x lRl+ N + N2 then, given a solution u of the modified equation, we set
n
-
2
Gl(x,t) = Gl(x,t,u,Du,D u)
to bring the equation to the form (3.20). The results formulated below are then applicable, provided assumptions on 0 1 are such that the resulting function G l satisfies (G1) (or (G1),) for any global solution u one wishes to consider. A condition like sup
(x,u,p,q)El1xlR 1+ N+N2
101(X,t,U,p,q)1 ~ Ce-,t
(t > 0)
189
would suffice. Another way in which the symmetric problem (3.1), (3.2) can be perturbed is allowing the domain D to depend on time and be asymptotically symmetric in a suitable sense. This can be covered by problems considered here to some extent. In case the variable domain was sufficiently smooth, by a time dependent change of coordinates one could transform the problem to a problem on a fixed symmetric domain and the equation in the transformed problem would be asymptotically symmetric. We consider classical solutions of (3.20), (3.21) satisfying (U1), (U2). We do not assume that they are nonnegative, rather we assume that they are asymptotically nonnegative in the sense that all elements of their limit sets are nonnegative. We first state an asymptotic symmetry theorem assuming that G l , G 2 decay as in (G1), not necessarily exponentially, but the solutions are assumed asymptotically (strictly) positive. Note that hypotheses (D2) is not needed in this theorem. Theorem 3.7. Assume (Dl), (Fl)-(F3), (GJ) and letu be a global solution of (3.20), (3.21) satisfying (Ul), (U2). Further assume that for each Z E w(u) one has Z > 0 on D. Then for each Z E w(u)
Z(-XbX') = Z(Xl'X')
«XbX') ED),
and Z strictly decreasing in Xl on Do = {x ED: in the form ZXl < 0 provided ZXl E C(Do).
Xl
> a}. The latter holds
The is theorem is proved in Ref. 35. Because of the assumption that all elements of w(u) are positive, this is not a generalization of Theorem 3.1. It is quite natural to ask whether the positivity assumption can be relaxed so as to only require that at least one element of w( u) be positive, as in Theorem 3.1. The answer is negative if G l , G 2 are merely assumed to decay as in (G1), or even if they decay exponentially as in (G1)-y with a small exponent I > O. See Ref. 35 for a counterexample. As the next theorem of Ref. 35 shows, the weaker asymptotic positivity condition is sufficient, provided (G1), holds with a sufficiently large I > O. Theorem 3.8. Assume (Dl), (D2), (Fl) - (F3). Then there exists I > 0 depending on ao, /3, N, D such that if (Gl)-y holds, the following is true. If u is a solution of (3.20), (3.21) satisfying (Ul), (U2), such that Z ~ 0 for each Z E w(u) and there exists ¢ E w(u) with ¢ > 0 on D, then for each Z E w(u)
Z(-XbX') = Z(Xl'X')
«Xl'X') ED),
190
and either z == 0 on n or z is strictly decreasing in holds in the form ZXI < 0 if ZXI E c(no).
Xl
on no. The latter
Problems with rotational asymptotic symmetry, such as asymptotically symmetric perturbations of problem (3.10), (3.11), can be treated in a usual way, applying the reflectional symmetry results after any rotation, see Ref. 35. An extension of the partial symmetry result of Theorem 3.2 to asymptotically symmetric problems can also be found in Ref. 35. 4. Quasilinear equations on ]RN
In this section we consider quasilinear parabolic equations on ]RN of the following form (using the summation convention as usual)
Ut = Aij(t,u, V'u)U XiXj
+ f(t,u, V'u),
E ]RN; t E J.
X
(4.1)
Similarly as for problems on bounded domains, we take either J = (0, 00) (and think of solutions of a Cauchy problem, not always indicating the initial condition explicitly) or J = (-00, T) for some T ~ 00. We assume the following conditions on the functions A ij , f : J x [0, 00) X ]RN ---) R
(Q1) Aij(t,u,p), f(t,u,p) are of class Cl in u and p = (Pl, ... ,PN) uniformly with respect to t. This means that A ij , f are continuous on J x [0,00) X ]RN together with their partial derivatives OuAij, ouf, 0pIAij,"" 0pNAij, op,!,"" 0pNf; and if h stands for any of these partial derivatives, then for each M > 0 one has lim
o:Su,v,JpJ,JqJ:SM, tEJ
Ih(t,u,p) - h(t,v,q)1 = O.
(4.2)
Ju-vJ+Jp-qJ--+o
(Q2) (Aijkj is locally uniformly elliptic in the following sense: for each M > 0 there is a~ > 0 such that
Aij(t, U,p)~i~j ~ a~I~12 (~=
(6, .. . ~N)
E ]RN, t E J, u E
[0, MJ, Ipi ~ M).
(4.3)
(Q3) f(t, 0, 0) = 0 (t E J) and there is a constant 'Y > 0 such that
ouf(t, 0, 0) < -'Y
(t E J).
(Q4) For each (t, u,p) E J x [0,00) X ]RN and i,j
Aij(t,u, Pop) = Aij(t,u,p),
= 1, ... , N
( 4.4)
one has
f(t,u,Pop) = f(t,u,p),
A lj == Ajl == 0 if j
:f. 1.
191
We consider global positive solutions of (4.1) satisfying the following boundedness and decay conditions:
u(x, t), lux.{x, t)l, IU XiXj (x, t)1 < do
(x E JRN, t E J),
(4.5)
where do is a positive constant, and Ix
llim sup{u(x, t), IU Xi (x, t)l, IU XiXj (x, t)1 : t E J, i, j = 1, ... ,N} = 0 (4.6) -><XJ
The above assumptions on Aij and f are suited for reflectional symmetry results (with respect to a hyperplane perpendicular to el = (1,0, ... ,0». We will also formulate rotational symmetry results, under additional assumptions. The assumptions in this section are the same as in Ref. 61,62. Note in particular, that the equation is quasilinear, rather than fully nonlinear as in the previous section. This is an essential requirement for the method used in Refs. 61,62. The fact that the nonlinearities are independent of x is not so important. A generalization to x-dependent problems under suitable monotonicity assumptions is not difficult, but the proofs would become a little cumbersome (see Section 4 of Ref. 61 for a discussion of the hypotheses and a comparison to elliptic problems on JRN). We further remark that if the functions Aij and f are slightly more regular (Holder continuous in t) then it is sufficient to assume the boundedness and decay of u and U Xi • For semilinear equations considered in the introduction it is sufficient to assume the boundedness and decay of u alone.
4.1. Solutions on (0, CX» Let J = (0,00). As in Section 3.1, the asymptotic symmetry of solutions u of (4.1) will be described in terms of the functions in the w-limit set of u. Assuming (4.5), (4.6), the orbit {u(·, t) : t 2: 1} is relatively compact in Co(JRN) the Banach space of all continuous functions on JRN decaying to 0 at Ixl = 00 (it is equipped with the supremum norm). Consequently, the w-limit set
w(u):=
{1>: 1> = limu(., tn)
for some tn
-->
oo},
(4.7)
with the limit in CO(JR N), is a nonempty compact subset of Co(JRN) and one has lim distcoORN) (u(·, t),w(u»
t->oo
= o.
We now state a theorem on asymptotic reflectional symmetry and its corollary on asymptotic radial symmetry of positive solutions of (4.5), (4.6) (the proofs are given in Ref. 61).
192
Theorem 4.1. Assume (Q1)-(Q4). Let u be a global positive solution of (4.1) satisfying (4.5) and (4.6). Then either u(·, t) ~ 0 in L<X:>(IRN) or else there exists A E IR such that for each ¢ E w(u) and each x in the halfspace IRf = {x : Xl > A} one has ¢(P>.x) = ¢(x),
OXl¢(X) <
(4.8)
o.
The next corollary concerns rotationally invariant equations of the form Ut
= A(t, u, lV'ul)~u
+ f(t, u, lV'ul).
Corollary 4.2. Let (Q1)-(Q3) hold, let Aij any i,j, and Aii(t,u,p)
= Au(t,u,q),
(4.9)
== 0 if i =I j, Au == Ajj for
f(t,u,p)
=
f(t,u,q),
whenever Ipi = Iql. Let u be a positive solution of (4.1) satisfying (4.5) and
e
(4.6). Then eitheru(·,t) ~ 0 in Loo(IRN) or else there exists E IRN such that for each ¢ E w(u) and each x, y E IRN with Iy - el = Ix - el > 0 one has one has ¢(x - e) = ¢(y -
e),
V'¢(x - e) . (x - e) <
o.
We emphasize that the hyperplane (or center) of symmetry is the same for all elements of w(u). In the case of radial symmetry, it in particular follows that, unless u decays to 0 as t ~ 00, all ¢ E w(u) have a unique point of maximum which is independent of ¢. This and the fact that u(-, t) approaches its w-limit set in a Cl sense (thanks to (4.5), (4.6») imply that for large t the function u(·, t) has a unique point of maximum which stabilizes (converges as t ~ 00) to the point E IRN. Of course, the solution itself may not stabilize.
e
4.2. Solutions on (-cx:>, T) and linearized problems Now we take J = (-00, T) for some T :::;: 00. The results in this subsection are taken from Ref. 62. We consider solutions of (4.1) satisfying (4.5) and (4.6). It can be proved (see Ref. 62) that with J = (-00, T), the decay condition (4.6) on u is satisfied if u is sufficiently small near Ix I = 00 (uniformly in t) and then the decay is necessarily exponential. We first formulate theorems on symmetry of positive solutions for all times.
193
Theorem 4.3. Assume (Q1}-(Q4) and let u be a positive solution of (4.1) on (-00, T) satisfying (4.5) and (4.6). Then there exist .A E lR such that for each t < T and x in the halfspace lRf = {x : Xl > .A} one has u(P>.x, t) = u(x, t),
(4.10)
aX! u(x, t) < O.
For equations in the rotationally invariant form (4.9) we have the following result.
Corollary 4.4. Let (Q 1)-(Q3) hold. Assume that Aij Ajj for any i, j, and Aii(t,u,p)
=
Aii(t,u,q),
f(t,u,p)
=
== 0 for i i= j,
Aii
==
f(t,u,q)
whenever Ipi = Iql· Let u be a positive solution of (4.1) on (-oo,T) satisfying (4.5) and (4.6). Then there exists ~ E lRN such that for each t < T and each x, y E lRN with Iy - ~I = Ix - ~I > 0 one has u(x, t) = u(y, t), V'u(x - ~,t) . (x - ~)
< O.
(4.11) (4.12)
Now assume u is a positive solution of (3.1) on (-00, T) as in Theorem 4.3. We examine symmetry properties of bounded solutions of the linearized equation Vt = aij(x, t)V XiXj
+ bi(x, t)V Xi + c(x, t)v,
x E lR N , t E (-00, T)
(4.13)
where aij(x, t)
= Aij(t, u(x, t), V'u(x, t)),
+ U XkXt (x, t)AkePi (t, u(x, t), V'u(x, t)), c(x, t) = fu(t, u(x, t), V'u(x, t)) + UXkXt(x, t)Aklu(t, u(x, t), V'u(x, t)).
bi(x, t) = fpi (t, u(x, t), V'u(x, t))
(4.14)
Theorem 4.5. Assume (Q1}-(Q4). Let u and .A be as in Theorem 4.3 and let v be a bounded solution of (4.13) on (-00, T). Then there exist a constant CI and a solution 'IjJ of (4.13) on (-00, T) such that 'IjJ(P>.x, t)
= 'IjJ(x, t)
(x E lR N , t < T),
(4.15)
and
(4.16)
194
If, in addition, lim (v(P,xx,t)-v(x,t))=O
t-t-oo
for some x in {x : Xl > .x}, then
CI
=
(4.17)
o.
The corresponding result concerning the radial symmetry reads as follows.
Theorem 4.6. Let the hypotheses of Corollary 3.3 be satisfied and let u and ~ be as in that corollary. Let v be a bounded solution of (4.13) on (-00, T). Then there exist constants CI, .•. , CN and a solution 'IjJ of (4.13) on (-00, T) such that x f---+ 'IjJ(x -~, t) is radially symmetric for each t < T and (4.18)
If, in addition, there is a point x with Xi > tE~()V(PJiX, t) - v(x, t))
~i'
i = 1, ... ,N, such that
= 0 (i = 1, ... , N),
(4.19)
where PJi is the reflection in the hyperplane {x : Xi = ~i}' then v(x -~, t) is radially symmetric for each t < T. Similarly as for problems on bounded domains, let us define the centerunstable space of the solution u as the space of all bounded solutions of the linearized equation (4.13) on (-00, T) and the unstable space as the space of all solutions v of (4.13) such that IIv(·, t)IILOO(IRN) -+ 0 as t -+ -00. The previous result says that the unstable space is spanned by radial solutions, and the center-unstable space is spanned by radial solutions and the spatial derivatives of u (which, of course, are not radially symmetric). This is a natural extension of the results on the symmetry of the center-unstable space of positive entire solutions of parabolic equations on bounded domains, as formulated in the previous section. If equation (4.1) is semilinear and autonomous, Ut = L).u + f(u), and the solution u is a positive steady state, Theorem 4.6 restates a well known result on the unstable space of the linearized Schrodinger operator (see Refs. 34,60, for example).
5. Cooperative systems We now give extensions of Theorems 3.1, 3.2 to cooperative systems. In this section n is a bounded domain in ]RN satisfying conditions (D1), (D2) of Section 3.
195
We consider the following Dirichlet problem for a system of nonlinear parabolic equations
OtUi = Fi(t,x,u,Dui,D2ui), Ui(X, t) = 0,
(x,t) ED x (0,00), i = 1, ... ,n, (5.1) (x, t) E oD x (0,00), i = 1, ... ,n. (5.2)
Here n 2': 1 is an integer, u:= (Ul,'" ,un), and for each i E S:= {1, ... , n}
Fi : (t,x,u,p,q)
1--+
Fi(t,x,u,p,q)
E jRn
is a function defined on [0,00) x n x Oi, where Oi is an open convex subset 2 of jRn+N+N invariant under the transformation Q: (u,p,q)
1--+
(U,-Pl,P2, ... ,PN,ij),
_ _ {-qi j qij
if exactly one of i, j equals 1,
%-
.
otherwIse.
Note that while system (5.1) is fully nonlinear, it is only weakly coupled in the sense that the arguments of Fi do not involve the derivatives of Uj for j -=I- i. We assume that F = (FI , ... , Fn) satisfies the following hypotheses: (N1) For each i E S the function Fi : [0,00) x n x 0i -+ jRn is continuous, differentiable with respect to q and Lipschitz continuous in (u, p, q) uniformly with respect to (x, t) E x jR+: there is (3 > such that
°
n
IFi(t,x,u,p,q) - Fi(t,x,u,p,ij)1 ::; (31(u,p,q) - (u,p,ij)1 ((x, t) En x jR+, (u,p, q), (u,p, ij) E Oi). (N2) There is a positive constant ao such that for all i E S, (t, x, u, p, q) E [0,00) x n x Oi, and ~ E jRN one has N
'L" &(t,x,u,p,q)~j~k OFi 2': ao 1~ 12 . j,k=l qjk (N3) For each i E S, (t,u,p,q) E [0,00) x Oi, and any (Xl,X' ), (Xl'X') E D with Xl > Xl 2': one has
°
Fi(t, ±XI, x', Q( U,p, q))
=
Fi(t, Xl, x', u,p, q) 2': Fi(t, Xl, x', u,p, q) .
(N4) For all i,j E S, i -=I- j, (t,x,u,p,q) E [0,00) x n x Oi one has
OFi VUj
~(t,x,u,p,q)
2': 0,
whenever the derivative exists.
In some results we need to complement (N4) with the following condition.
196
(N5) There exists a > 0 such that for any nonempty subsets I, J with In J = 0, I u J = 5 there exist i E I, j E J such that
c
5
BPi
-B (t, x, u,p, q) ~ a Uj
for all (t, x, u,p, q) E [0,00) x
n x 0i such the derivative exists.
Hypotheses (Nl)-(N3) are analogous to those assumed in the scalar case in Section 3. Condition (N4) characterizes the cooperativity structure of the system and (N5) is a form of an irreducibility condition. Note that the derivatives in (N4) and (N5) exist almost everywhere by (Nl). A model problem is the cooperative reaction-diffusion system (1.14) considered in the introduction. As in the scalar case, we consider classical solutions of (5.1), (5.2) satisfying the following two conditions sup maxllui(" t)IILOO(O) tE[O,co) 'ES
lim h-+O
sup
x,xEn, t,~l,
< 00,
lu(x, t) - u(x, 01 =
(5.3)
o.
(5.4)
[t-f[,[x-x[
The orbit {u(.,t) : t ~ l} of a solution satisfying (5.3), (5.4) is relatively compact in the space E = (C(n))n and then the w-limit set
w(u)
= {z: Z = k-+co lim U(·,tk)
for sometk
-+
oo},
is nonempty, compact in E and it attracts u(·, t) as t -+ 00. By a nonnegative solution, we mean a solution which is nonnegative in all its components. The following result on asymptotic symmetry of nonnegative solutions is proved in Ref. 36. Theorem 5.1. Assume (Di), (D2), (Ni) - (N4). Let u be a nonnegative solution of (5.1), (5.2) satisfying (5.3) and (5.4). Assume in addition that
one of the following conditions holds:
(i) there exists
Z
=
(Zl"'" zn) E w(u) and i E 5, the function
Zi
is even in
Xl:
(5.5)
197
°
and either Zi == on 0 or Zi is strictly decreasing in holds in the form (Zi)Xl < if (Zi)Xl E C(Oo).
°
Xl
on 0 0 . The latter
It can be proved (see Ref. 36) that if (N5) holds, in addition to all the other hypotheses of Theorem 5.1, then for any
°
°
liminf Fi(t,x,O,O,O)
t.-oo,xEO
~
0,
(5.6)
then the following holds for any a nonnegative solution u of (5.1), (5.2) satisfying (5.3) and (5.4): for each i E Seither !!Ui(-, t)IILOO(o) -+ or else there exists
°
°
Theorem 5.2. Let (Dl), (Nl) - (NS) hold and assume that 0), is connected for each>' E (0,£). Let u be a nonnegative solution of (5.1), (5.2) satisfying (5.3) and (5.4). Then there exists>. E [0, £) such that for all Z = (Zl,' .. ' Zn) E w(u) \ {OJ and i E S one has
Zi(XI,X') = zi(2). - Xl,X') and Zi is strictly decreasing in (Zi)Xl < if (Zi)Xl E C(O),).
°
Xl
((Xl,X') EO),)
on 0),. The latter holds in the form
A more general version of this theorem is proved in Ref. 36. 6. On the proofs: a comparison of bounded and unbounded domains In this section we describe, on a rather general level, the methods used in the proofs of some of the above symmetry results. Doing so, we want to highlight key differences in the proofs of theorems dealing with bounded domains and those for ]RN. For simplicity, we consider spatially homogeneous equations Ut
= F(t,u, Du, D 2 u),
X
E
O,t
E J,
(6.1)
198
where 0 is either a bounded domain as in Section 3 (it satisfies (D1), (D2)) or 0 = JRN, and J = (0,00) or J = (-00, T). If 0 is bounded, we couple (6.1) with Dirichlet boundary condition (3.1). In the case 0 = JRN, we restrict the class of admissible equations to quasilinear ones, as in Section 4, and we assume the hypotheses (Q1)-(Q4) of that section to be satisfied. Likewise, for equations on bounded domains we assume the hypotheses (F1 )-(F3), of Section 3 to be satisfied. Consider first the case J = (0,00). Assume that a nonnegative solution u satisfies (U1), (U2) (if 0 is bounded) or (4.5), (4.6) (if 0 = JRN), and that the w-limit set w(u) of u contains a strictly positive element
°
(we are using the notation introduced in Section 2). Note that for each fixed A, (S».. is equivalent to the following statement
Vv ~
°in 0>.. for all
Z
E
w(u).
(6.2)
As the first step of the proof one shows that (S)>.. holds if A < £, A ;: : : £ (recall that £ = 00 if 0 = JRN). Having proved that, one sets
AO := inf{ll: (S)>.. holds for each A E [Il,£)}.
In view of (6.2), the definition of AO clearly implies that each Z E w(u) is monotone nonincreasing in Xl > AO (by additional arguments, not to be recalled here, one can show that Z is actually strictly decreasing unless it is identical to 0). Also, by continuity, V>"oz ~ 0, in 0>"0' To prove the symmetry of each Z E w(u) about H>..o, one needs the opposite inequality: V>"oz :::; in 0>"0' To prove it, set
°
AO := sup{1l : (S-».. holds for each A E (-£, III
},
where (8-».. is the statement
with O~ := {x EO: Xl < A} (we only take A E (-£,0] if 0 is bounded). The desired symmetry follows from the second key step which consists in
199
°
showing that AO = AO if n = ]RN and that AO = if n is bounded (analogous arguments then also give AO = 0). The technical background of the above steps comprises several estimates of solutions of linear parabolic problems of the form
Vt
= at(x, t)V XiXj
+ bt(x, t)VXi + c>'(x, t)v,
v 2: 0,
x En>., t > 0,
(6.3)
x E 8n>., t > 0.
(6.4)
Using Hadamard formulas and the assumptions on (5.1), one shows that the function v = V>. u = u>' - u satisfies such a linear problem with bounded measurable coefficients (they are bounded by /3, the a Lipschitz constant of F with respect to (u,p, q)) and with uniformly elliptic leading part (aij )f,j=l (the ellipticity constant coincides with that of (5.1)). To carry out the first step of the above scheme, i.e., to show that (8)>. holds for A !=:j i!, in case n is bounded, one applies a parabolic version of the maximum principle on small domains. It says, in essence, that if n>. has sufficiently small measure (which is the case if A !=:j i!), then any solution v of (6.3) satisfies
[[v-(" t)[[Loo(OA) ::; 2e- k (t-r) [[v-(-, 7)[[Loo(OA)
(t > 7 > 0),
(6.5)
°
where the exponent k > can be taken as large as desired, upon making [n>.[ smaller (see Ref. 63 for the proof; similar results appeared earlier in Refs. 4,6 and the elliptic predecessor can be found in Ref. 11). If n = ]RN then n>. is a halfspace and the small domain maximum principle does not help in the first step. Instead, assumption (4.4) is used which implies that the coefficient c>' in (6.3) is less than -1/2 if A !=:j 00 and this gives (6.5) with k = -1/2. Note that this time k cannot be taken arbitrarily large. This point is not so relevant here, but it is in the second step. In the second step, first with n bounded, it is to be shown that AO = 0. The proof is by contradiction. Assuming AO > 0, one first utilizes the definition of AO and the assumption that there is a positive element ¢; E w( u) to conclude that v := V>'o u has the following property. There is a sequence tn ----4 00 such that for any Dec n>.o lim inf v(x, t n )
xED,n~(X)
> 0.
(6.6)
Also, since (8ho holds,
[[v-(-,t)[[Loo(OAO)
----4
°as t
----4
00.
(6.7)
The contradiction consists in showing that (6.7) remains valid if v is replaced with v := V>.U, where):' < AO is sufficiently close to AO (this means
200
that (Sh holds for all ); ::::; AQ sufficiently close to AQ, in contradiction to the definition of AQ). The argument goes as follows. Take Dec n>.o such that n>.o \ D has small measure, hence also In x\ DI is small for ); ~ AQ. By the equicontinuity assumption on u, (6.6) remains valid if v is replaced with v = VXu, where); < AQ is sufficiently close to AQ. Also liv-(" tn)IILOO(n x) can be assumed small for such );, by (6.7). Now, v(·, t) remains positive in D for t in some interval [tn, T). Take the maximal T with that property. Two estimates on v are next derived. In the first one, one shows that liv-(-, t)IILOO(n;.\D) decays exponentially in t E [tn, T) with an exponent that can be assumed as large as desired, upon adjusting D and taking); closer to AQ, if necessary. This follows by an application of the small domain maximum principle to (6.3), (6.4), with A = ); and with n>. replaced with nX \15. The second estimate concerns the function sUPxED v(·, t) for t E [tn' T). While it can decay exponentially as well, one can control its decay by an exponential function with a fixed exponent (the exponent is determined by ellipticity constant of (6.3) and a bound on its coefficient - quantities independent of A). Hence, for t E [tn,T) and); ~ AQ, Ilv-(-, t)IILOO(nx\D) decays much faster than sUPxED v(·, t). The two estimates are next combined, via an extension of the Harnack inequality which applies to sign-changing solutions. The resulting estimate shows that v(.,t) = v+(-,t) - v-(·,t) remains positive in D for all times, i.e., T = 00. In particular, the exponential decay estimate on v- remains valid for all t > tn which gives the desired contradiction. The details of the above proof are given in Ref. 63. We remark that the arguments involving the two kinds of exponential estimates are not needed if u(x, t) is assumed to stay away from zero for all x E n (this is in particular true when u is a steady state). In that case the proof is considerably simpler (see Appendix A of Ref. 63). When carrying out the second step of the above scheme for n = lR N , i.e., when ruling out the possibility that Ai) < AQ, one does not have the luxury of using the small domain maximum principle (although a maximum principle for narrow domains does playa role). Still, there are some similarities with the above arguments, in particular, two exponential estimates of v = VXu, with); < AQ, ); ~ AQ, are derived in the proof, one in a large bounded domain D cc and another one in D (see Ref. 61). The key difference is that in the latter exponential estimate one cannot assume the exponent to be arbitrarily large, at best one can assume it to be close to the constant, from assumption (4.4). Therefore the first estimate has to be done much more carefully: one wants to control sUPxED v(·, t) from below
lRfo
lRf \
201
by an exponential function with a sufficiently small exponent. This can be done invoking a perturbation argument, relying on the behavior of V,x. u for A ;::: AD. However, the perturbation argument gives the desired result only if the leading coefficients aij in (6.3) are independent of A. This is where the quasilinear structure of the equation is used. The scheme for proving the symmetry results for J = (-00, T) is similar, except one considers the following statement in place of (S),x.. (T),x.
(V,x.u) (x, t) ;:::
°
(x
E n,x., t E
(-00, T)).
Although there are differences at several technical steps, key arguments of the proof for J = (0,00) can be easily adapted. When dealing with asymptotically symmetric equations, such as those considered in Section 3.3, additional technical difficulties stem from the fact that the equation corresponding to (6.3) has an additional term, say f(x, t), which makes it a linear nonhomogeneous equation. One needs to quantify the exponential decay that has to be assumed on f for the exponential estimates on the solutions to go through. Alexandrov-Krylov estimate (see Theorem III.3.9 of Ref. 48 or Theorem VII.7.1 of Ref. 53), which is the parabolic extension of Alexandrov-Bakelman-Pucci estimate (see Ref. 18 for example), plays an important role in the proof (see Ref. 35).
7. Applications and some open problems As mentioned in the introduction, once asymptotic symmetry of positive solutions of a parabolic problem is established, it can be very useful in the study of the temporal asymptotic behavior the solutions. To illustrate this, we consider the following nonautonomous reaction-diffusion problem on the unit ball in JR.N: Ut
u(x, t)
= D.u
= 0,
+ f(t, u),
x E n := B(O, 1), t x E 8n, t > 0.
> 0,
(7.1) (7.2)
We assume that f : JR. x [0,00) --+ JR. is a CI-function which is time-periodic with period T > 0: f (t +T, u) == f (t, u). We consider a nonnegative bounded solution u of (7.1), (7.2). By standard parabolic estimates, such a solution is also bounded in CI(fl) hence it satisfies the equicontinuity condition (U2) of Section 3. Therefore, by Corollary 3.3, u is asymptotically radially symmetric. The question is what can be said about the asymptotic behavior of the solution in relation to the time-periodic structure of the solution. The following theorem answers that question.
202 Theorem 7.1. Under the above assumptions, let u be a nonnegative bounded solution of (7.1), (7.2) . Then there exists a solution p(x, t) of (7.1), (7.2) which is r-periodic in t: p(." + r) == p, and such that
lim lIu(" t) - p(., t)IILOO(o) ~ 0.
t->(X)
Thus u approaches, as t ~ 00, a r-periodic solution of (7.1), (7.2). Such convergence results are true in one-space dimension in a more general setting and without any symmetry structure (see Ref. 14). However, in higher space dimension the convergence does not take place in general and the symmetry is crucial (see for example Refs. 65,66 and references therein). Theorem 7.1 is proved in Ref. 23 (a similar convergence result in the autonomous case can be found in Ref. 43). Let us indicate how symmetry results are used in the proof. We already know, by the asymptotic symmetry, that all elements of w( u) are radial functions. An invariance principle for time-periodic equations says that w(u) consists of entire solutions of (7.1), (7.2). Since these are radial solutions, they can be viewed as solutions of the one-dimensional problem Wt = Wrr
N -1
+ - - Wr + f(t, w)
wr(O, t) = w(l, t)
r
= 0,
= 0,
r E (0,1), t E JR.,
(7.3)
t ER
(7.4)
Moreover, it follows from other general properties of w-limit sets, that the entire solutions forming w(u) belong to what is called the chain recurrent set of the periodic-parabolic problem (7.3), (7.4). Analysis of this one-dimensional problem based on intersection comparison arguments (see Refs. 22,23) reveals that the chain recurrent set consists of r-periodic solutions. Once it is known that w(u) consists of r-periodic solutions, one can apply known convergence criteria tailored to such a situation. 13 ,42 These give a sufficient condition, formulated in terms of the linearization along solutions in w( u), for w( u) to actually consists of just one periodic solution. The sufficient condition can be verified using a version of the symmetry result for the linearized equation given by Theorem 3.5, see 23 for details. In connection with the above sketch, let us mention an interesting open problem. What makes the proof of Theorem 7.1 rather complicated is that even though we know that the positive solution u is asymptotically symmetric, we do not know if its asymptotic behavior is determined by a solution of the radial problem. This is only obtained at the end, after a rather involved analysis of the radial problem. Had it been known that u symmetrizes with an asymptotic symmetric phase, that is, that Ilu(" t) - z(·, t)IILoo ~ for
°
203 some symmetric solution z, the convergence of u would immediately follow from known convergence results for radial solutions. Whether such an asymptotic phase exists in general and, if not, what are sufficient conditions for its existence is a very interesting problem that has not been addressed. Another, probably closely related, problem concerns the rate of symmetrization, that is, the rate with which a positive solution converges to the space of symmetric functions. For equations on bounded domains, sufficient conditions for the rate to be exponential are given in Ref. 6. It appears that for problems JRN the exponential rate can be established as well, although no detailed proof is available yet. From other applications of symmetry results and related ideas, let us mention Liouville-type theorems. Those assert, that for a specific class of elliptic or parabolic equations on the whole space or a subspace there are no nontrivial nonnegative solutions. A well-known example is the Liouville theorem for the elliptic equation t1u
+ uP =
°
N
x E lR ,
where p is a sub critical Sobolev sub critical exponent: 1 < p < (N +2)/(N2)+ (see Refs. 21,39). Chen and Li's proof of this result, see Ref. 21, uses symmetry: they show that that any hypothetical positive solution would have to be radially symmetric around any origin in lRN. This implies that no such solution can exist. In parabolic equations, the scenario symmetrythen-nonexistence has also been shown to work in some situations. For example, in Ref. 64 the following problem on the halfspace lRb" is considered under some conditions on the exponent p > 1. Ut - t1u
= uP,
u=O,
x E lRf/, t E lR,
x E f)lRf/, t E lR,
Expanding on an idea of Ref. 29, a Liouville theorem for this problem is proved, after first establishing the monotonicity in Xl of each hypothetical positive solution. This is but an example of symmetry results for a parabolic equation on an unbounded proper subdomain of lRN. Such equations have not been studied systematically from the symmetry point of view. References 1. A. Aftalion, J. Busca, and W. Reichel. Approximate radial symmetry for overdetermined boundary value problems. Adv. Differential Equations, 4:907932, 1999. 2. A. D. Alexandrov. A characteristic property of spheres. Ann. Math. Pum Appl., 58:303-354, 1962.
204 3. H. Amann. Global existence for semilinear parabolic systems. J. Reine Angew. Math., 360:47-83, 1985. 4. A. V. Babin. Symmetrization properties of parabolic equations in symmetric domains. J. Dynam. Differential Equations, 6:639-658, 1994. 5. A. V. Babin. Symmetry of instabilities for scalar equations in symmetric domains. J. Differential Equations, 123:122-152, 1995. 6. A. V. Babin and G. R. Sell. Attractors of non-autonomous parabolic equations and their symmetry properties. J. Differential Equations, 160:1-50, 2000. 7. H. Berestycki. Qualitative properties of positive solutions of elliptic equations. In Partial differential equations (Praha, 1998), volume 406 of Chapman & Hall/CRC Res. Notes Math., pages 34-44. Chapman & Hall/CRC, Boca Raton, FL, 2000. 8. H. Berestycki, L. Caffarelli, and L. Nirenberg. Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:69-94 (1998), 1997. Dedicated to Ennio De Giorgi. 9. H. Berestycki, L. A. Caffarelli, and L. Nirenberg. Symmetry for elliptic equations in a half space. In Boundary value problems for partial differential equations and applications, volume 29 of RMA Res. Notes Appl. Math., pages 27-42. Masson, Paris, 1993. 10. H. Berestycki, F. Hamel, and R. Monneau. One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J., 103:375396,2000. 11. H. Berestycki and L. Nirenberg. On the method of moving planes and the sliding method. Bal. Soc. Brasil. Mat. (N.S.), 22:1-37, 1991. 12. F. Brock. Continuous rearrangement and symmetry of solutions of elliptic problems. Proc. Indian Acad. Sci. Math. Sci., 110:157-204, 2000. 13. P. Brunovsky and P. Polatik. On the local structure of w-limit sets of maps. Z. Angew. Math. Phys., 48, 1997. 14. P. Brunovsky, P. Polai:ik, and B. Sandstede. Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal., 18:209-215, 1992. 15. J. Busca, M.-A. Jendoubi, and P. Polacik. Convergence to equilibrium for semilinear parabolic problems in ]RN. Comm. Partial Differential Equations, 27:1793-1814,2002. 16. J. Busca and B. Sirakov. Symmetry results for semilinear elliptic systems in the whole space. J. Differential Equations, 163:41-56, 2000. 17. J. Busca and B. Sirakov. Harnack type estimates for nonlinear elliptic systems and applications. Ann. Inst. H. Poincare Anal. Non Lineaire, 21(5):543-590, 2004. 18. X. Cabre. On the Alexandroff-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math., 48:539-570, 1995. 19. A. Castro and R. Shivaji. Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. Comm. Partial Differential Equations, 14: 1091-1100, 1989.
205 20. G. Cerami. Symmetry breaking for a class of semilinear elliptic problems. J. Nonlinear Anal. TMA, 10:1-14, 1986. 21. W. X. Chen and C. Li. Classification of solutions of some nonlinear elliptic equations. Duke Math. J., 63(3}:615-622, 1991. 22. X.-Y. Chen and P. PohiCik. Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations. J. Dynam. Differential Equations, 7:73-107, 1995. 23. X.- Y. Chen and P. Polac:ik. Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball. J. Reine Angew. Math., 472:17-51, 1996. 24. Y. S. Choi and A. W. Shaker. On symmetry in semi-linear elliptic systems with nonnegative solutions. Appl. Anal., 53:125-134, 1994. 25. B. Chow and R. Gulliver. Aleksandrov reflection and geometric evolution of hypersurfaces. Comm. Anal. Geom., 9(2}:261-280, 2001. 26. C. Cortazar, M. del Pino, and M. Elgueta. The problem of uniqueness of the limit in a semilinear heat equation. Comm. Partial Differential Equations, 24:2147-2172, 1999. 27. F. Da Lio and B. Sirakov. Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations. J. European Math. Soc., 9:317-330, 2007. 28. E. N. Dancer. On nonradially symmetric bifurcation. J. London Math. Soc., 20:287-292, 1979. 29. E. N. Dancer. Some notes on the method of moving planes. Bull. Austral. Math. Soc., 46:425-434, 1992. 30. E. N. Dancer and P. Hess. The symmetry of positive solutions of periodicparabolic problems. J. Comput. Appl. Math., 52:81-89, 1994. 31. D. G. de Figueiredo. Monotonicity and symmetry of solutions of elliptic systems in general domains. NoDEA Nonlinear Differential Equations Appl., 1:119-123,1994. 32. D. G. De Figueiredo and J. Yang. Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal., 33(3):211-234, 1998. 33. Y. Du. Order structure and topological methods in nonlinear partial differential equations. Vol. 1, volume 2 of Series in Partial Differential Equations and Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. 34. E. Feireisl and H. Petzeltova. Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differential Integral Equations, 10:181-196, 1997. 35. J. FOldes and P. Polacik. On asymptotically symmetric parabolic equations. in preparation. 36. J. Foldes and P. Polacik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Disc. Cont. Dynamical Systems, to appear. 37. B. Gidas, W.-M. Ni, and L. Nirenberg. Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68:209-243, 1979. 38. B. Gidas, W.-M. Ni, and L. Nirenberg. Symmetry of positive solutions of nonlinear elliptic equations in ]Rn. In Mathematical analysis and applications,
206 Part A, pages 369-402. Academic Press, New York, 1981. 39. B. Gidas and J. Spruck. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 34:525-598, 1981. 40. C. Gui, W.-M. Ni, and X. Wang. On the stability and instability of positive steady states of a semilinear heat equation in Rn. Comm. Pure Appl. Math., 45:1153-1181, 1992. 41. C. Gui, W.-M. Ni, and X. Wang. Further study on a nonlinear heat equation. J. Differential Equations, 169:588-613, 2001. 42. J. K. Hale and G. Raugel. Convergence in gradient-like systems with applications to PDE. J. Applied Math. Physics (ZAMP), 43:63-124, 1992. 43. A. Haraux and P. Polacik. Convergence to a positive equilibrium for some nonlinear evolution equations in a ball. Acta Math. Univ. Comenian. (N.S.), 61:129-141,1992. 44. P. Hess and P. PolaCik. Symmetry and convergence properties for nonnegative solutions of nonautonomous reaction-diffusion problems. Proc. Roy. Soc. Edinburgh Sect. A, 124:573-587, 1994. 45. C.K.R.T. Jones. Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions. Rocky Mountain J. Math., 13:355-364, 1983. 46. B. Kawohl. Symmetrization-or how to prove symmetry of solutions to a PDE. In Partial differential equations (Praha, 1998), volume 406 of Chapman & Hall/CRC Res. Notes Math., pages 214-229. Chapman & Hall/CRC, Boca Raton, FL, 2000. 47. B. Kawohl and G. Sweers. Inheritance of symmetry for positive solutions of semi linear elliptic boundary value problems. Ann. Inst. H. Poincare Anal. Non Lineaire, 19:705-714, 2002. 48. N. V. Krylov. Nonlinear elliptic and parabolic equations of the second order, volume 7 of Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. Russian Original: "Nauka", Moscow 1985. 49. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Urall'ceva. Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968. Russian Original: "Nauka", Moscow 1967. 50. C. Li. Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains. Comm. Partial Differential Equations, 16:491526, 1991. 51. C. Li. Monotonicity and symmetry of solutions offully nonlinear elliptic equations on unbounded domains. Comm. Partial Differential Equations, 16:585615, 1991. 52. Y. Li and W.-M. Ni. Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm. Partial Differential Equations, 18:1043-1054, 1993. 53. G. M. Lieberman. Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, NJ, 1996. 54. C.-S. Lin and W.-M. Ni. A counterexample to the nodal domain conjecture and a related semilinear equation. Proc. Amer. Math. Soc., 102:271-277, 1988.
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209
TRAVELLING PULSE WAVES NON-VANISHING AT INFINITY FOR THE DERIVATIVE NONLINEAR SCHRODINGER EQUATION Kouya Imamura and Kunimochi Sakamoto
1. Introduction and main result
We study the following derivative nonlinear Schrodinger equation:
(t, x)
E
JR. x JR.,
(1.1)
where u(t, x) is a {::-valued function of (t, x). This equation comes from a model equation describing planar Alfven waves propagating in one direction (say, x-axis). Briefly, the derivation goes as follows ([15] for detail). The motions of electrons and ions in plasmas are described by a two-fluid model. This model consists of the equations of continuity and the equations of motion which are coupled with the Maxwell's equations. After a number of simp lying assumptions, this model is further reduced to a one-fluid model. If one looks for planar waves in the latter model, perturbative expansions around a trivial state give rise to equations of the magnetic field
After appropriate rescaling of dependent and independent variables, the equations for p and q are then given by
+ qxx + ((p2 + q2)p)x = qt - Pxx + ((p2 + q2)q)x =
Pt {
0,
0,
(1.2)
which are obtained as consistency conditions in the perturbative expansions. If we set u = p + iq, then (1.2) is equivalent to (1.1). Therefore, we refer to both (1.1) and (1.2) the derivative nonlinear Schrodinger equation, and use them interchangeably according to our convenience.
210
The initial value problem for (1.1) has been extensively studied by many authors [5,8, 11-13J; local and global well-posedness results for (1.1) have been established on Sobolev spaces HS(JR) (8 > 1/2). As a special class of solutions that fits into the Sobolev space setting, it is known ([1, 4, 14]) that (1.1) has a two-parameter family of solitary waves Uw,c(t,x) defined by
Uw,c(t, x) = eiwt Rw,c(x - ct) exp{i8 w ,c(x - ct)},
(vS-H)
2
where parameters w > 0, c E JR satisfy c < 4w. The amplitude Rw,c and the phase 8 w ,c are explicitly given by
Rw,c(x) =
Vw { [4w::: c2 coshh/4w -
8 w,c(x) =
2c x - 43 IX
-00
c2 x) - 2Jw
}] -1/2 '
2
IRw,c(1])1 d1].
The obital stability (cf. [1,3,4]) of Uw,c(t, x) has been partially determined in [4J and completely established in [1], based upon an abstract framework developed in [3]. It is clear that these solutions decay exponentially to zero as Ixl ---+ 00, and hence they belong to HS(JR) (8 ~ 0). On the other hand, (1.1) admits solutions of other types, such as constant solutions (for any Zo E C, u(t,x) == Zo is a solution of (1.1)), periodic wave trains u(t,x) = rei(kx-wt) (with r > O,k E JR,w = k 2 + rk), solutions connecting a constant solution to another, and bounded non-decaying (as Ixl ---+ 00) solutions in general. Investigations into these types of solutions have been rather neglected. Taking into account the origin of (1.1) as compatibility conditions in the perturbative expansion, it is no doubt important to study these latter types of solutions, even though they may not belong to the energy or Sobolev spaces. One notices that the solitary waves in (vS-H) are of the form u(t, x) = eiwtv(t, x), and that the equation for v(t,x) is given by
(1.1-w) As opposed to (1.1), the only constant solution for (1.1-w) is v == 0. Due to the introduction of the term -wv, the two dimensional continuum of constant solutions of (1.1), except for u = 0, have been eliminated by the forceful time dependent rotation u ......., eiwtv. It is then the non-degeneracy of the isolated constant solution v = of (1.1-w) for w > that supports the existence of solitary waves in (vS-H) which decays to as Ixl ---+ ±oo. In other words, rotations induce the non-degeneracy of the trivial state u = 0.
°
°
°
211
In this paper, we will show that there is another mechanism which makes non-zero constant states non-degenerate. It is to exploit the translational symmetry of (1.1) along the spatial direction, and hence, we are particularly concerned with travelling waves of (1.1). The equation for travelling waves of (1.1) with speed c, u(t,x) = ¢(x - ct), is given by (1.3) where ~ = x - ct is a moving coordinate, and ¢(~) is called a wave profile. The object of the present paper is to find solutions of (1.3) satisfying (~->
±oo),
and to study their spectral properties. The equation (1.1) has symmetries as described in the following proposition. This fact will play important roles to investigate the solution structure of (1.1). Proposition 1. Let u(t,x) be a solution of (1.1). Then for each () E JR, X> 0 and (to,xo) E JR x JR,
and Vtr(t, x) := u(t - to, x - xo) are also a solution of (1.1). The proof is straightforward. Thanks to Proposition 1, without loss of generality, we may assume 0: = -1, that is,
¢(O
--+
-1
(~->
±oo).
(1.4)
In fact, if ¢(~) is a wave profile with speed c satisfying (1.4), then the wave profile ¢(~) satisfying lime ..... oo ¢(~) = 0: E (-00,0) has wave speed c = 10:1 2 c and is expressed as
(1.5) The existence of travelling waves is summarized as follows. Theorem 1. The equation (1.1) has a travelling wave solution satisfying
e.lim . . ±oo ¢(~)
= -1,
212
if and only if 1 < c < 3. In this case, there exist two families of travelling waves ¢s(~) and ¢L(~) which are explicitly represented as follows. ¢s(~)
= ps(~) + iqs(~),
¢d~) = PL(~)
where
1{
ps(~)=-l+-
4
qs(~) and
pd~) =
+ iqd~),
} c - 1 - 2 v1n2(1-Ktanh2(L~))2 2) (v~ 2 -c+1, 1 + Mtanh (L~)
2J2(3 - c)Mtanh(LO , (1 + M tanh2(L~))2 cosh2(L~)
=
-1 + 1-4 {(Vc=l + 2V2)2 (1tanh~(L~))2 _ c + I}, 1 + M-l tanh
qL(~) = _
1
K-
(L~)
2J2(3 - c)M- tanh(L~) . (1 + M-l tanh2(L~))2 cosh2(L~) 1
Parameters K, Land M are related to the wave speed c as follows. K = (V2 - ve=-r)(2V2 + ve=-r)
(V2 + ve=-r)(2V2 - ve=-r)' L =
21 J (c -1)(3 -
c),
M_V2-ve=-r - V2+ ve=-r' We exhibit the orbits of ¢s and ¢L, and ~-profiles of PS,qS,PL, and qL in Figs. 1, 2, and 3. The phase portrait in Fig. 1 indicates that the usual solution ansatz ¢(~) = r(~)eie(O is not appropriate to obtain these solutions, if not impossible. Note that:
0< K < 1, 0< M < 1,
o< L
1
::;
2'
lim K = 0,
c--+3
lim M = 0,
c--+3
lim K = 1,
c--+ 1
lim M = 1,
c--+l
lim L = lim L = O.
c--+3
c--+ 1
213
~
_ _~0.5
-0.5
0.5
1
1.5
cPL(~)
for c = 1.4.
-0.5
Fig. 1.
The orbits of cPs (0 and
4
Fig. 2.
The graphs of ps(~) and PL(O for c = 1.4.
Next, we discuss spectral properties of the travelling wave solutions described in Theorem 1. Let us note that (1.2) is equivalent to
+J(p) -c(p) +{V'ii(p,q)}~=O, ( p) q t q EE q E
(1.6)
214
Fig. 3.
The graphs of qs(~) and qdO for c = 1.4.
where
J= ( 0
-1
We linearize (1.6) around (p, q) = (PS,L, qS,L) to obtain the following eigenvalue problem: (1.7) where
and 2 -
R(O := \7 H(PS,L, qS,L)
-
(Hessian of H).
In order to study the spectral problem for (1. 7), we introduce the notion of essential spectrum. Note that there is considerable divergence in the literature concerning the definition of essential spectrum. We first recall the definition of Fredholm operator ([7, Chapter IV §5]). A closed linear operator T : X -> Y between Banach spaces X and Y is Fredholm, if R(T) is closed and dimN(T) < 00, codimR(T) < 00. For a Fredholm operator T, its index ind(T) is defined by ind(T) = dimN(T) - codimR(T).
215
Definition 1. Let T : X space X into itself.
--t
X be a closed linear operator on a Banach
1. A E C belongs to resolvent set p(T) if and only if A + T is Fredholm with ind(A + T) = 0 and N(A + T) = {a}. 2. A E C is in pure point spectrum 0" pp (T) if and only if A + T is Fredholm with ind(A +T) = o,N(A +T) -:J. {O} and A is isolated in C\p(T). 3. A E C is in essential spectrum O"ess(T) if and only if A+T is not Fredholm or A + T is Fredholm with ind(A + T) -:J. o.
Remark 1. We cannot necessarily conclude that
O"(T) := C \ p(T) = O"pp(T) U O"ess(T) is true in our definition (cf. comments after Proposition 2, §3). A generally accepted definition of essential spectrum is such that the formula above holds, i.e., the essential spectrum is the complement (in the spectrum) of pure point spectrum. Essencial spectrum O"ess(T) by our definition is fairly large. Another definition (see Kato [7]) of O"ess(T) is "the complementary set of the semiFredholm domain", where T is said to be semi-Fredholm if R(T) is closed and at least one of dimN(T) and codimR(T) is finite.
Theorem 2. For (1. 7), let us consider'c : L2(lR, (:2) we have
--t
L2(lR, (:2). Then
(i) O"ess('c) = ilR C (:. (ii) O"(,c) = O"ess('c) U O"pp(,c). (iii) ,C generates a CO-group of linear operators e- tC (t E lR) on LP(lR, (:2) (1 ::; p ::; 00). The identification of the essential spectrum O"ess('c) is rather easily accompliched, since it is determined by the asymptotic (I~I --t 00) states of the wave profile. On the other hand, the location of pure point spectrum O"pp('c) depends on the entire wave profile. When O"pp('c) = 0 in Theorem 2, the wave solutions in Theorem 1 are called spectrally stable. However, this spectral stability does not necessarily imply that these waves are orbitally stable (nonlinear stability). From Theorem 2 (iii), one may try to prove the local well-posedness of (1.1) near the travelling waves by using a standard method. Unfortunately, the usual semilinear approach fails for this case because of the so called derivative loss. In existing literature ([5, 8, 11]), this is overcome by
216
performing a nonlinar transformation called a gauge transformation. This method, however, does not work in our situation. 2. Existence of travelling pulse waves
Integration of (1.3) with respect to .; on (-00,';) gives
and we have ¢/ Let ¢(.;) = p(';) have p'
+ i([¢[2
+ iq(';) (p,q
- c)¢
= i(c -
1).
(2.1)
E JR) and substitute it into (2.1). We then
+ iq' + i(p2 + q2 _ c)(p + iq) = i(c _
- de '
1)
(' _ d)
that is,
p' = (p2 + q2 _ c)q, { q' = _(p2 + q2 _ c)p + c - 1.
(2.2)
This is a Hamiltonian system with a Hamiltonian function 1
H(p, qj c) = '4(p2 Fixed points of (2.2) are (p, q) o:±
+ q2)(p2 + q2 - 2c) =
(c - l)p.
(-1,0), (CL, 0), (0:+,0), where 1±y'4c-3 2
= --'--:,----
Since we are looking for solutions of (2.1), namely (2.2), which satisfy the conditions (1.4), the fixed point (p,q) = (-1,0) has to be a saddle point. The linearization of (2.2) around (p,q) = (-1,0) is
(:;)=(C~3 l~C)(:). The fixed point (-1,0) is a hyperbolic saddle if and only if the determinant of the matrix
is negative, that is,
1 < c < 3.
217
Next, we linearize (2.1) around «L,O) and (a+,O). We have
(p'q' ) = (0 c - 3a~
° c) (p) ( ° a~°- c) .
Let A± be eigenvalues of
a~ -
q'
c- 3a~
Since its determinant (c - a~)(c - 3a~) is positive (because of 1 < c < 3), A± are purely imaginary, and therefore, the fixed points (a±, 0) are centers.
2.1. Homoclinic orbits We notice that the wave profile ¢(O with lime ....t±oo ¢(t;) = -1 corresponds to an orbit of (2.2) homoclinic to (p,q) = (-1,0). Therefore we first try to represent this homoclinic orbit as q = q(p). Since (2.2) is a Hamiltonian system, its orbits are contained in the contour-lines of H(p, q). Therefore, we need to find the contour-line of H which passes (p,q) = (-1,0). The contour-line is determined by H(p, q) - H( -1,0) = 0. Since
H(-I,O) = 4(H(p,q) - H(-I,O)) =
3
c
-4 + 2'
°implies
q4 _ 2(c _ p2)q2
+ p4 - 2cp2 - 4(c - l)p + 3 -
2c = 0.
This is a quadratic equation in q2, and hence we obtain
(f(p) := 4(c - l)p + (c - 1)(c + 3)). This suggests that there may exist two homo clinic orbits which are biasymptotic to (p,q) = (-1,0) as t; ~ ±oo. It turns out that there do exist two homo clinic orbits. To show this, we first determine the range of p for which f(p) ~ and (c - p2) ± f(p) ~ 0. By elementary computations, we have
°
J
°J f(p) ~ ° ~
c - p2 { f(p) ~
Po '5 p '5 {3,
where
c+3
Po:= --4-' and c - p2 + { f(p)~O
{3:= 1- J2(c -1),
J f (p) ~ ° ~ p
< P < "" 0_
-
/'
218
where 'Y:= 1 + y'2(c-1).
Since Po < -1 < sented by
f3 <
'Y, there are two homo clinic orbits which are repre-
q~=c-p~-y'f(ps)
q'i =
{c -
PL - y' f(pL) + y' f(PL)
c - PL
(-1
(2.3)
+ iqLI ~ vic) + iqLI > vic)
(Po ~ P ~ -1), (Po ~ P ~ 1').
(2.4)
On the complex plane, we represent them as
+ iqs, + iqL.
2.2. Small and Large pulse waves ¢S,L In this section, we represent PS,L and qS,L explicitly as a function of ~. To obtain PS,L, we substitute (2.3) and (2.4) into the first equation of (2.2). The resulting equations are, respectively,
pi = _y' f(p)(c - p2 - y' f(p))1/2 { p(O) = f3
(2.5)
and
p' = -y' f(p)(c - p2 { p(O) = 1'. Changing variables by U as
= y' f(p), UI
+ y' f(p))1/2
(2.6)
(2.5) and (2.6) are, respectively, written
= -~y'9-(U) 2
(2.7)
and
(2.8) where
g±(U) := - U 4 + 2(c - l)(c + 3)U 2 ± 16(c - 1)2U + 16c(c - 1)2 - (c - 1)2(c + 3)2.
219
It is easy to see that the transformation U 1---4 -U converts (2.8) into (2.7). Therefore, we only need to solve (2.7) with U(O) = Ufj for Ps and with U(O) = U, for PL, where
Ufj = -(c - 1) + 2J2(c - 1), U, = -(c - 1) - 2J2(c - 1). Now, (2.7) reduces to
_~ =
21
U
dU
U{3,-y
(U -
C
+ I)J(U -
U,)(Ufj - U)'
The right-hand side is integrated by elementary computations and gives rise to the expressions of PS,L in Theorem 1. Once p(~) is known, the corresponding q(~) is obtained from Uf(~)
q(~) = - 2(c - 1)" 3. Spectral property of pulse wave solutions In this section, we prove Theorem 2. First, we prepare some propositions to prove Theorem 2. Proposition 2. (Kato [7, Chapter IV §5]) Let
r
r c
C be defined by
:= {AI A + C is not Fredholm}.
Then, c-r consists of at most countably many open connected components, say {OJ}~o, so that
(a) ind(A + C) = mj E Z for all A E OJ; (b) for each j, there is a set of isolated exceptional points Aj,k, numbers Vj, "'j E Z with Vj - "'j = mj and rj,k with 0 < rj,k < 00 such that
"'j
• dimN(A + C) = Vj, codim'R.(A + C) = for Ai- Aj,k . • dimN(Aj,k + C) = Vj + rj,k, codim'R.(Aj,k + C) = + rj,k.
"'j
By using Proposition 2, let us elaborate on our definitions of essential spectrum and pure point spectrum. We define 0 p by Op =
U{Oj I mj
= 0, Vj = "'j =
O}.
j
Then Op \ {exceptional points} is exactly the resolvent set p(C) and exceptional points in 0 p are pure point spectra. The essential spectrum is given
220
by
O"ess(.c) =
r u U{Oj I mj i= O}. j
It is just a logical consequence that the remainder
C - O"ess(.c) - O"pp(.c) = U{Oj I mj = 0, Vj
i- O}
j
may be non-empty. This set is contained in O"(.c), and therefore, the statement (ii) of Theorem 2 is not trivial. If we let
then (1.7) is equivalent to d
d~ ~ = A(~; .\)~
(3.1)
where
A(~;'\)
=
02 ( .\J + JR'(~)
I2
JR(~)
- cJ
)
.
Let us define asymptotic matrices
A±oo(.\) =
~
lim ...... ±oo
A(~;'\)
= (
o o
O~
.\
-.\
o
1
o o c-3
Definition 2. For a hyperbolic square matrix A, the Morse index of A, i(A), is the dimension of the generalized eigenspaces of A corresponding to eigenvalues with positive real part.
Let A.~ : BC 1 (JR,C 4 )
-+
BC(JR,C4 ) be defined by
A\ ~
d
= d~ ~ - A(~; .\)~.
Proposition 3. (Henry [6], Palmer [9, 10])
(i) .\ + .c is Fredholm if and only if A\ is Fredholm, and ind(.\ + .c) = ind(A>.).
221
(ii) AA is Fredholm if and only if asymptotic matrices A±oo(A) are both hyperbolic, and
where i(A±oo(A)) is the Morse index of A±oo(A).
Since A-oo(A) = A+oo(A) in our case, we automatically have ind(A C) = as soon as A + C is Fredholm.
°
+
Lemma 1.
(i) Aoo(A) is non-hyperbolic if and only if A E ilR. (ii) For ~A =I- 0, i(Aoo(A)) is two. (iii) For each 0 =I- ±n/2, there exists ro = ro(O) > such that
°
AE p(C)
if argA = 0,
IAI
~
roo
Proof (i) Let J.L denote eigenvalues of Aoo(A)j
J.L4 - (c - 1)(3 - c)J.L2 - 2(2 - C)AJ.L + A2 = O.
(3.2)
If Aoo(A) is non-hyperbolic, (3.2) must have a purely imaginary root, that is, J.L = iv (v E JR). Substituting it into (3.2), we have
v 4 + (c -1)(3 - c)v 2 - 2i(2 - C)AV + A2 = 0. Let A = Al have
+ iA2
(AI, A2 E JR) and separating real and imaginary parts, we
v4 + (c - 1)(3 - c)v 2 + 2(2 - C)A2V + AI - A~ = 0, { -(2 - C)AIV + AIA2 = 0.
(3.3)
If Al =I- 0, the second equation in (3.3) gives A2 = (2 - c)v. We substitute it into the first equation in (3.3) to get
v
4
+ v 2 + Ai = 0.
Namely, v = Al = 0. This contradicts Al =I- 0. Therefore, A E iR On the other hand, if Al = 0 the second equation in (3.3) is trivially satisfied and the first equation in (3.3) has at least two (counting with multiplicity) real roots for any A2 E R Therefore, (3.2) has pure imaginary roots if A E HR. (ii) We first restrict A to be real. For A = 0, (3.2) has a double root at J.L = and simple real roots at J.L = ±J(c - 1)(3 - c). When real A is perturbed from zero, the two simple roots remain as real simple roots. On the other
°
222
hand, the double root at /L = 0 splits into two real roots as >. E lR \ {O} is near zero. In fact, for /L ~ 0 (3.2) is approximated by
-(c - 1)(3 - c)/L2 - 2(2 - c)>'/L + >.2 = (>. + (c - 1)/L)(>' + (c - 3)/L) = 0 since /L 4 is of higher order. Therefore (3.2) for>. E lR \ {O} near zero has two real roots /L~
>. 1-c
--
and
>. 3-c
/L~--.
These two roots have opposite signs for any real >. E lR \ {O} near zero. For large 1>'1 (>. E lR), (3.2) may have non-real zeros. However, they cannot cross the imaginary axis due to statement (i). Now, statement (ii) remains true even if>. is non-real, as long as >. moves in left or right complex half plane, again, due to the statement (i). (iii) Let us rewrite (3.1) as follows.
(:)~ =v1-\I(~) v1-\I (~) ~ =>'J (:) + JR'(~) (:) + v1-\I (JR(~) We rescale ~ = JfXI~ and divide the resulting equation by get
cJ)
(~).
JfXI and 1>'1 to
(:)'=(~),
(if.)' = e 'Ij;
where'
ill J
(
= d/d{ By letting 1>'1
--400,
(3.4)
we obtain (3.5)
which defines an invertible operator
for () =/:. ±7T /2. In fact, the coefficient matrix of (3.5) is hyperbolic if () =/:. ±7T/2 and the real parts of its eigenvalues are at least cos( I~I + %) away from the imaginary axis for I()I < ~. By the robustness ([6]) of hyperbolicity (exponential dichotomy), (3.4) has an exponential dichotomy on lR
223 for sufficiently large IAI, say,
IAI»
[COS2C~I +~)]-1
Interpreted in terms of the original equation (3.1), this means that A + £ defines an invertible operator for A = IAleiO, if IAI > 0 is sufficiently large and e -1= ±7f/2. 0 Proof of Theorem 2. (i) From Proposition 3 and the statement following it, A E a ess (£) if and only if Aoo(A) is non-hyperbolic. Therefore, Lemma l-(i) implies a ess (£) = ilR. (ii) For A E C - iJR, ind(A + £) = 0, by Lemma l-(i),(ii) and Proposition 3-(ii). On the other hand, Lemma l-(iii) implies that A in an open subset of C - iJR belongs to pee). Therefore, from Proposition 2, the exceptional points in C - iJR has to be isolated eigenvalues of £ with finite multipicity. (iii) From a general theory of generation of Co-groups ([2]), it suffices to show IIA(A + £)-111 ~ M for A E JR with IAllarge, where M > 0 is a constant. If (3.4) is written as <J>' = A(e; A)<J>, then for 1 ~ p ~ 00 the equation (A + £)u = hE U(JR, C 2 ) is written as
- -- - N1( -Jh(~) 0) .
U' - A(~; A)U
=
(3.6)
From the proof of Lemma 1, the left hand side of (3.6) has an exponential dichotomy on JR, and hence IlullLP ~ MllhilLp/IAI for some M > 0 follows 0 ([6]), proving IIA(A + £)-111 ~ M. References 1. M. Colin and M. Ohta, Stability of solitary waves for derivative nonlinear Schrodinger equation, Ann. 1. H. Poincare - AN. 23 (2006) 753-764. 2. J. A. Goldstein, Semigroups of Linear Operatros and Applications, Oxford University Press, 1985. 3. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987) 160-197. 4. B. Guo and Y. Wu, Orbital stability of solitary waves for the nonlinear derivative Schrodinger equation, J. Differential Equations 123 (1995) 35-55. 5. N. Hayashi, The initial value problem for the derivative nonlinear Schrodinger equation in the energy space, Nonlinear Anal. 20 (1993) 823-833. 6. D. Henry, Geometric Theory of semilinear parabolic equations, Lecture Notes in Math. Vol 840, Springer-Verlag, New York-Berlin-Heidelberg, 1981. 7. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, BerlinHeidelberg-New York, 1966.
224
8. T. Ozawa, On the nonlinear Schr6dinger equations of derivative type, Indiana Univ. Math. J. 45 (1996) 137-163. 9. K. Palmer, Exponential Dichotomies and Transversal Homoclinic Points, J. Differential Equations 55 (1984) 225-256. 10. K. Palmer, Exponential Dichotomies and Fredholm Operators, Proc. Amer. Math. Society 104 (1988) 149-156. 11. S. Tan and L. Zhang, On a weak solution of the mixed nonlinear Schr6dinger equations, J. Math. Anal. 182 (1994) 409-421. 12. M. Tsutsumi and 1. Fukuda, On solutions of the derivative nonline Schr6dinger equation. Existence and uniqueness theorem, Funkcialaj Ekvacioj 23 (1980) 259-277. 13. M. Tsutsumi and 1. Fukuda, On solutions of the derivative nonline Schr6dinger equation II, Funkcialaj Ekvacioj 24 (1981) 85-94. 14. W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized comlex Ginzberg-Landau equation, Physica D 56 (1992) 303-367. 15. M. Wadati, H. Sanuki, K. Konno and Y. Ichikawa, Circular polarized nonlinear Alfven waves - A new type of nonlinear evolution equation in plasma physics, Rocky Mountain J. Math. 8 Vol. 1 and 2, Winter and Spring (1978) 323-331.
225
THRESHOLD AND REDISTANCING DYNAMICS FOR GEOMETRIC MOTIONS Selim Esedoglu Department of Mathematics, University of Michigan. Ann Arbor, MI48109. Email: [email protected]. Research supported in part by NSF grant DMS-0605714. Steven J. Ruuth Department of Mathematics, Simon F'raser University. Burnaby, British Columbia, V5A 1S6, Canada. Email: [email protected]. Research supported in part by a grant from NSERC Canada. Richard Tsai Department of Mathematics, University of Texas at Austin, TX 78712. Email: [email protected]. Research supported in part by NSF under agreement No. DMS-0513394 A class of algorithms for computing high order geometric motions of planar curves is developed. The algorithms alternate between two simple steps-a convolution step and a "thresholding" step--to evolve planar curves according to combinations of Willmore flow, surface diffusion flow, and more general curvature dependent flows. When a closed curve is represented by a level set function and the thresholding step is replaced by redistancing the level set function according to its zero level contour, the proposed algorithm provide efficient and accurate computations of these geometric motions on uniform grids. Keywords: Threshold dynamics, mean curvature motion, Willmore flow, surface diffusion.
1. Introduction
Higher order geometric motions of curves and surfaces arise in a variety of applications. They constitute a difficult class of computational problems, due to the stiffness in the discretized systems and the potential development of topological singularities during the curve motions. These motions typically correspond to the steepest descent for energies that are defined over curves and surfaces. For example, one of the algorithms proposed in
226
this paper computes gradient descent for the energy E(C) :=
L
al<;,2
+ /3ds
(1)
where C is a closed curve in the plane, I<;, is its curvature, ds is the length element, and a, /3 are two positive constants. Energy (1) is sometimes referred to as Euler's elastica energy. The Willmore functional in two dimensions is given by Ew(C) =
L
~
1<;,2
ds.
(2)
Gradient descent for (2) moves the curve C with normal velocity given by
W =
-1<;,88 -
1
-I<;,
2
3
•
(3)
See [1] for the derivation. The motion resulting from (2) moves every point on the curve C by a normal velocity given by Vn
=
a( -21<;,88
-
1<;,3)
+ /31<;,.
In the motion by surface diffusion of a curve, the normal velocity is given by
This motion can be derived as the gradient flow of E in H- 1 with a = O. Computer vision, digital image processing, and materials science are among the application areas that would greatly benefit from improvements in computational methods for higher order geometric motion of curves and surfaces. Indeed, many important problems of image processing and computer vision involve minimizing curvature dependent functionals over curves and surfaces. For instance, the active contours (or "snakes") model [2] of Kass, Witkin, and Terzopoulos originally calls for minimizing an energy that includes the square of the curvature integrated along the curve. Another important example is the segmentation with depth involving disocclusion [3,4] model of Nitzberg, Mumford, and Shiota [5,6], where curvature dependent functionals are to be minimized in order to extract information about the three dimensional arrangement of objects making up a scene from a single two dimensional image of the scene. Yet another related problem of the field that leads to higher order geometric flows is the image inpainting problem of Bertalmio et. al., where the goal is to repair damaged regions in an image by connecting level lines of intensity using smooth curves [7].
227 A few techniques are being applied and investigated for the type of high order geometric motions under consideration. Explicit, semi-implicit, and implicit surface algorithms have been introduced to track the evolution of the curves and surfaces, see for example [8-14J. Due to the high order derivatives involved in the motions, stability conditions impose a severe efficiency drawback for explicit discretization methods. On the other hand, the nonlinear nature of the motions makes it difficult to develop implicit schemes; questions remain about whether usable implicit schemes perform quite as efficiently as what is required in applications. In this paper, we will consider algorithms for computing a number of these geometric motions, and illustrate them on some of the mentioned applications. Our approach is based on the threshold dynamics (or diffusion generated motion) idea of Merriman, Bence, and Osher [15,16J proposed originally for motion by mean curvature. MBO schemes have excellent stability properties. We believe this makes them particularly promising for rapid solution of problems in which steady states are sought; see [17J and [18J for an example in segmentation with depth problem of image processing. Defining the interface to be the boundary of a region ~ C lR N, the idea of [15,16J is to alternate the following two steps: First, convolve the characteristic function of ~ with a Gaussian kernel. Then, threshold the resulting function at ~. More precisely, their algorithm can be described as follows: Set ~o = ~. Alternate the following steps for k = 0,1,2, ... in order to generate the sets ~k so that a~k approximate the motion of a~o by mean curvature at the discrete times tk = k· t::..t: (1) Form the convolution:
K(x)
:= (lI;k
* Gt::..t)(x)
where Gt::..t(x) is the fundamental solution of the heat equation in N space dimensions. (2) Set:
~k+l
= {x
ElRN :
K(x):::::
~}.
A major benefit of the algorithm is that both steps are extremely simple and can be implemented very efficiently: The convolution can be done, for example, via the fast Fourier transform, and the second step - the thresholding - is trivial. A potential drawback of this algorithm is that the accuracy on a fixed grid is poor since there is no sub cell resolution during the thresholding step. This limitation can be overcome using Fast Fourier
228
transforms on grids adapted to the evolving interface [20]. The original MBO algorithm that we described above has been rigorously shown to converge to motion by mean curvature of a~ in [21,22]. Several generalizations of the basic algorithm are given in [23-28] and some applications to image segmentation are given in [29,30]. Grzibovskis and Heintz [31], propose a generalization of the MBO algorithm to Willmore flow for two dimensional surfaces in three space, which is a fourth order evolution of the interface obtained as gradient descent for the Willmore functional. We concentrate on two dimensions since this is the particularly relevant setting for many applications, especially in image processing. We further introduce a new algorithm based on diffusing the signed distance functions to the curves and redistancing nonlinear functions of these "diffused" functions. Again, the structure of this new algorithm can be conveniently described as follows: (1) Form the convolution:
Kj(x):= (d~k
* Gt:.tJ(x),j =
1,2,···,1
(4)
where G Llt J (x) is the fundamental solution of the heat equation in N space dimensions. (2) Set: ~k+1 = {x E]RN : 9(K1 (x),K2 (x),··· , Kt(x)) ~
where
9 is a function from]Rt
I-t
O},
(5)
R
Perform redistancing [19] to create
d-~k+l.
2. Expansions, Truncations, and Thresholding Let ~ C ]R2 denote the set whose boundary is to be evolved according to a desired motion law. Set ~o := ~ to be the initial curve. This section describes how one can generate the approximations ~k to the flow at the discrete times tk = k . t:.t by using diffusion of either the characteristics fucntions or signed distance functions of ~k • Consider a smooth interface which is given by the graph of the function g(x), initially passing through the origin, where it is tangent to the x-axis, so that g(O) = 0 and g'(O) = o. We shall work with this curve given by the graph of g(x) and the region of the plane that lies below this curve in the xy-plane, namely {(x, y) : y < g(x)}; our interface - the curve - is the boundary of this region. At the origin we have gss = gxx which implies that
229
Thus the normal velocity of a Willmore flow can be expressed as 5
W =
-gxxxx
+ "2'"
3
(6)
at the orgin of this coordinate system.
2.1. Expansions Near the Curve This section summarizes a convolution-thresholding algorithm for Willmore flow in two dimensions [18J. The idea is to consider the convolution of the characteristic function of the interior of the curve (the region {(x, y) : y < g(x)}) with a suitable kernel. First, define the function F(x, y, t) to be the convolution of the characteristic function of the region below our curve with the fundamental solution of the heat equation at time t:
F(x,y,t) =
00 100 1 _ -00 -00 41rte 1
(x-0 2 +(Y-'7)2 4'
l{(I;,1/):g(I;»1/}(~,'f})d~d'f}.
(7)
To see the asymptotics of the convolution on the curve, we evaluate F(x, y, t) at x = 0 and along the y-axis and reduces (7) to
F(O
t) =-1 - -1" 2 2y'tif y
00 1 1 4tn -00 _
1 Y
e-"( 2 /4t
0
1 9
e-t;2/4t
d'"V+ I
(8)
(1;) e-(Y-1/)2/4td'f}~.
0
Expanding the exponentials around 0, and replacing g(~) by the first few terms of its Taylor series at 0, and noting that we are interested in F(O, y, t) at y = O(t 2), t -+ 0, we obtain 1 yC 1/ 2 + __ 1 g(2)t 1/ 2 + __ 1 g(4)t 3/ 2_ F(O y t) =-1 - __
"
2
2y7i
2y7i
8~ (g(2») 3 t 3/ 2 + o(t5/ 2),
4y7i
(9)
where g(j) is the j-th derivative of g. In terms of the curvature", = g(2) and the Willmore flow speed W (see equation (6)) at x = 0 for the curve given by the graph of g(x), the last equation reads:
!-
_1_yc 1/ 2 + _1_"'tl/2 _ _ 1_ Wt3 / 2 + O(t5/ 2). (10) 2y7i 2y7i 4y7i We may thus use different values of t and consequently F, in combina-
F(O, y, t) =
2
tions of suitable algebraic procedures to produce a new function whose 1/2 level set is vnt distance away from the origin. This will be equvalent of an Euler step moving the origin in the y-direction with a speed V n .
230
2.2. Trancations and Thresholding We start with an example which illustrates our new thresholding idea in more detail. The result will be used in later parts of the paper. From the expansion derived above, consider the function 1
A(x,y):= ve(2Llt)"4F(x,y,.j2Llt/O) 1
(2Llt) ~
ve
F(x,y,O.j2Llt).
(11)
(ve -
If we threshold A(x,y) at (2';t) "4 l/ve), the resulting curve would intersect the y-axis at a y value such that: _1_
2y'1f
(~ _ 0
0) y _ 4y'1f 2Llt (~_ 0) W + O(M~) = 0
O.
The above threshold value was chosen so that constant terms of order o( ~d) drop out. Simplifying, we see that the solution is: 3
Y = LltW + O(Llt'i) First of all, note that y = O(Llt) = O(t2), as required by our expansion. Second, we see that the curve moves with the following normal speed:
Vn=W during this time step of size Llt, with error O( y't;i).
2.2.1. Willmore with Lower Order Terms Consider interface motions with normal velocities of the form:
Vn = co(x, y, t)
+ Cl(X, y, t)", + W.
(12)
Velocities of this type arise e.g. in computer vision; more specifically, in the segmentation with depth model of Nitzberg, Mumford, and Shiota [5], as well as in digital image inpainting models inspired by that work [7,32]' it is necessary to minimize an energy of the form Ie a+ (1/2)(3",2 da over curves C, which leads to the normal velocity v = a", + (3W for a curve evolving via gradient descent for that energy. We consider combinations of the form
aF(x, y, .j2Llt/O) + (3F(x, y, O.j2Llt)
+ 'Y.
Again, the idea is to take the correct linear combination between two different Gaussians to form the convolution kernel. After some cauculations, we arrive at the following algorithm:
231
(1) Form the combination:
C(x,y) = ~t
1
e- O)co(x, y, t) 1
A(x, y) - ~t(2~t)-4 Cl (x, y, t)B(x, y) - 2y7rC
where A(x, y) is as in (11) and B(x, y) is as in (13). (2) Set:
Ek+l =
{(X,y) E JR2 : C(x,y) 2:
(2~t)! (v'e -1/v'e)}
Here, the function B is defined by
B(x, z)
:=
O-! (F(X, y, v'2~t/0) -
~) -
O! (F(X, y, 0v'2~t) -
~).
(13)
Thus we see that thresholding the function C above by the value
(2~t)! (v'e - 1/v'e) 2
induces instantaneous normal velocities of the form (12), with error O( v;:::t).
2.2.2. Surface Diffusion If we break away from simple linear combinations of convolutions, we can also obtain thresholding algorithms for other important fourth order flows. Here, we extend our results to motion by surface Laplacian of curvature for a curve, which is an important geometric motion that arises in a number of applications. The normal velocity takes the form
where s is the arclength and K, is the curvature of the curve. Using the functions A and B defined in (11) and (13), we arrive at
A(O, y)
1
7f
+ (1/0 _ 0)2 (2~t)4 B 1
3
(2~t)! r;; r;; (0, y) - - 2-(v 0 - l/vO)
= 2y7r (1/0 - 0) {y -
(14) M·
3
K,ss}
+ O(M2).
This leads to the following algorithm for motion of a curve by surface diffusion:
232 (1) Form the combination:
D(x,y) := A(O,y)
7r ( + (1/()-()2 2M ) 14B 3(O,y)
where- A(x, y) is as in (11) and B(x, y) is as in (13). (2) Set:
~k+l = {(X,y) E R2
: D(x,y)
~ (2~t)i (VB -l/VB)}
This algorithm differs from the standard convolution generated motions in that the first step - i.e. the convolution step - involves a nonlinear expression. Nevertheless, the both steps of the algorithm maintain the efficiency of their counterparts in the standard MBO scheme, since the nonlinear operation involved in the convolution step is pointwise.
2.3. Signed Distance Function to a Smooth Curve and Redistancing The procedures described in the previous subsections can be generalized to achieve high sub cell resolution on uniform Cartesian grids, while keeping the simplicity of linear diffusions. One way to do this is to replace the charactersitic function of the set enclosed by ~ by a signed distance function of it. As above, we consider ~ locally as the graph of a function g, and replace the characteristic function l{(~,1);g(O>1)}in (7) by a signed distance function, d, to the graph of g, and the thresholding step by a redistancing (or level set reinitialization) step [19,33]. In order to derive the formula that moves the interface at the right speed, we need to perform similar calculations as in the previous subsection, starting from (7) and (8), but using the chosen signed distance function. The first thing is to obtain a suitable expansion of the distance function d. In [34], the authors show that near d = 0, d is very close to a straight line in the direction normal to the interface. In particular, many derivatives in the normal direction vanish. In two dimensions, the derivatives of d in the direction perpendicular to 'V d can be related to the curvature and its derivatives of the level sets. Using these facts, we can compute an asymptotic expansion of d about an arbitrary point on the interface and use it to understand how linear diffusion moves the zero level set of d, i.e. the interface. More specifically, consider a smooth closed curve r and a Cartesian coordinate system centered at a point on r such that r is locally the graph
233 of the function g, satisfying g(O) = g'(O) = 0, and g"(O) = -11;(0). Denote by d the signed distance function to the curve r, such that d(x, y) > 0 if y > g(x), and d(x, y) < 0 if y < g(x). We assume that g is sufficiently smooth such that d is smooth and satisfies the eikonal equation
!V'd!
= 1,d(x,g(x» = o.
Let G t:..t denote the heat kernel, the effect of diffusing d for b.t amount of time is then [34] (15) assuming that y = O(b.t). The expansion can be computed in the same fashion as shown in the previous subsection. We refer the readers to [34] for more detail. We first see that thresholding G t:..t * d at zero yields higher order truncation error, O( b.t 2 ), in the evolution, since the zero level set of G t:..t * d is lI;b.t + O(b.t 2 ) away from that of d. This is actually not surprising as we recall that the mean curvature of the level sets of d can be evaluated by
V'd 1I;=-V'·!V'd!· Since !V'd! as
=
1, we have k
= -b.d.
Forward Euler method is then written
As in the previous subsections, one can devise more elaborate linear combinations of kernels in order to cancel the leading order error terms (Le. Richardson extrapolation) and thus obtain thresholding algorithms with higher order accuracy. Let us review this idea in the setting of Richardson's extrapolation. For any given function u, we shall use S to denote the redistance operator such that S[u] creates the signed distance function d satisfying {±d < O} = {±u < O} and {d = O} = {u = O}. Then for convenience, define
and denote
234
From the discussion above, we can estimate the "local error" related to the scheme: d
(2)
-(2)
d
2
= dI;(to+At)
i:::1t + AT + O(i:::1t 3 ),
= dI;(to+At)
+ Ai:::1t2 + O(i:::1t3 ),
where E(to + i:::1t) is the interface moving under curvature flow at time to + i:::1t. Consequently, as in a standard Richardson's extrapolation,
approximates the curvature flow with a smaller local error. This leads to the following high order scheme for mean curvature motion:
(1) Diffuse and form the intermediate distance functions:
d(l)
= Rat2 [do], d(2) = Rat2 [d1l
(2) Redistance:
The next location of the interface is now defined by
{(x,y): dk+l(X,y) =
OJ.
2.3.1. General Curvature Motions Let d(x, y) be the signed distance function to the interface. We want to move the zero level set of d by a function of curvature f (K,) via a threshold dynamics strategy. Recall our expansion of a diffused signed distance function around its zero level set:
for y
= O( i:::1t).
The curvature
K,
can be approximated by
GAt
*d -d i:::1t
235 Hence, if we look at the function
u(x, y)
= d - I ( Gt..t*d-d) b.t b.t,
we see that along the y-axis,
u(O, y) = y - 1(K,)b.t + O(b.t 2 ). Thus, thresholding u(x, y) at the speed of I(K,).
°will move the interface asymptotically at
3. Numerical Implementations and Experiments The previous section presents asymptotic analyses of diffusion on charactersitic functions or signed distance functions of a set whose boundaries are the curves to be moved at a desired velocity. The formulas derived there reveal what function and how thresholding should be performed. This section demonstrates some of the computational results obtained by our implementation. We first briefly discuss our numerical implementations for the convolution step. Convolution with a Gaussian kernel eliminates the high frequencies that are present in the initial characteristic function. Since this is a linear convolution, the different modes do not interact and there is never a need to approximate the high frequency modes when using the Gaussian kernel. This idea was exploited in [20] to give an efficient spectral discretization of the MBO method, and in [31] to give an efficient implementation of their algorithm for Willmore flow. However, analogues of the algorithm using other radially symmetric kernels can also be obtained. Another generalization of the MBO scheme to general convolution kernels with convergence result can be found in [35]. Another way to achieve good computational efficiency and accuracy in the convolution step is to replace the convolutions with the Gaussian kernel by solving the heat equation using standard 5-point finite difference stencils. This procedure has been studied extensively in the scientific computing communities and optimal methods such as multigrid methods have been widely implemented as the state-of-the-art procedure for linear diffusion. In the algorithms for Willmore flow or surface diffusion, various values of the parameter e can be chosen, but notice that e > 1 is required for the consistency of the resulting algorithm. This implies that the combined kernel used in the convolution step of the algorithm will be negative sufficiently far away from the origin. The negativity discourages mergers (topological
236
changes) from taking place as two interfaces approach one another. Interestingly, this behavior agrees with the underlying Willmore flow since the Willmore functional (2) becomes infinite at corners and cusps. In the following examples, except the ones involving redistancing, FFTs on unequally spaced grids are used to carry out the spatial discretization [20]. These adaptive methods lead to results which are essentially free of spatial discretization error, allowing us to better focus on the behavior of the underlying semi-discrete method. Typical values for the adaptive spatial discretization were 256 x 256 basis functions and a finest grid cell size of 2- 14 • For our time step choices, this gives kernel sizes that are much larger than the finest grid cell size.
3.1. Expanding Circle For our first example, consider the evolution of a circle according to Willmore flow. By symmetry, the curve will remain a circle and will evolve according to the ordinary differential equation
R= ~
(~r
From this differential equation, it is straightforward to determine the radius at any time t:
R(t) = ytR(0)4
+ 2t.
Our tests set the initial radius equal to 0.2 and evolve for a time T = 0.0025. In this and subsequent examples, the time step-size was chosen sufficiently fine to give a clear indication of the convergence rate. This leads to the absolute errors and numerically observed convergence rates given in Table 1 (all measured in terms of area). The expected convergence rate of O(...;t;i) is clearly observed. Table 1. Absolute errors in the final area of the expanding circle. l!..t
T/256 T/512 T/I024 T/2048 T/4096
Error
Conv. Rate
0.01918 0.01277 0.00864 0.00595 0.00413
0.587 0.562 0.540 0.527
-
Remark: In this tests set, () = (1.001)2. Ignoring numerical cancellation issues, values of () close to 1 are expected to produce better accuracy since
237
such values correspond to kernels with smaller support. Our next example examines this choice more closely.
3.2. Combined Curvature and Willmore Motion To examine the numerical behavior of the general algorithm, we consider the motion law Vn
= -10+7rK,+2W
and approximate the initial numerical velocity at the origin for the polynomial interface (16)
Taking () = (1.001)2 and varying flt gives the relative errors and numerically observed convergence rates displayed in Table 2. The expected convergence rate of O( v;:;:t) is clearly observed. Table 2. Relative errors in initial velocity for the evolving polynomial, and a general motion.
flt T/800 T/1600 T/3200 T/6400 T/12800 T/25600
Error in velocity
Conv. Rate
0.0575 0.0204 0.0148 0.0107 0.0077 0.0055
1.50 0.46 0.47 0.48 0.49
-
3.3. Surface Diffusion Flow Figure 1 shows a result of a test that evolves an initial ellipse until it becomes nearly circular. The results obtained are clearly stable in the sense that there are no spurious oscillations, even with very large time steps that far exceed the CFL condition of explicit schemes. We also find that the relative error in the area enclosed by the final curve is less than 1.5%. (The exact value of the final area is known from the fact that surface diffusion flow preserves the area enclosed by curves.) In fact, arbitrarily large time steps lead to regular evolutions and no observed instabilities. A further example of surface diffusion flow, this time with a more interesting, flower shaped curve as initial data, is given in Figure 2; compare to a similar computation in [36]. Figure 3 shows computations with the same
238
Fig. 1. Ellipse evolving under surface diffusion flow. A: Initial ellipse (t = 0). B: Intermediate contour (t = 0.0003125). C: Final contour approximates a circle (t = 0.0025). The time step was taken to be At = 0.0025/512.
o Fig. 2. A flower shaped curve evolving under surface diffusion flow. Upper left: Initial curve (t = 0). Upper right: Computed solution at t = 0.001/256. Lower left: Computed solution at t = 0.003/256. Lower right: Final contour is very closely a circle (t = 0.04/256). Time step-size was At = 0.001/(50 x 256).
initial data but with much larger choices of time step-size. As can be seen, the method is free of numerical instabilities even when the time step is so
239 large that the first plotted stage of the previous calculation is reached in a single step. This unconditional stability property was observed in all the motions we investigated.
Fig. 3. Illustration of the stability properties of the proposed surface diffusion algorithm: Arbitrarily large time steps can be taken, accuracy considerations being the only constraint. The curves, from left to right, represent the computed solution at a fixed time, starting from the same initial data as in the experiment of Figure 2, using 50, 5, and 1 time steps.
3.4. Shape Reconstruction
In [17], it was shown that threshold dynamics can be applied to minimize the energy (1) and thereby reconstruct missing (or occluded) parts of shapes described by their characteristic functions. That paper evolved the shape by sequentially carrying out a step of Willmore flow, followed by a step of the Merriman, Bence, Osher algorithm for curvature motion. Using the approach for general flows described in Section 2.1, these two steps can be combined into one. The corresponding simplified algorithm takes the following form: Let D be the image region, iJ be a subdomain of D where image (i.e. shape) information is known to be missing and ~o c D denote the initial guess for the shape. By alternating the following steps, approximations ~k, k = 0,1,2, ... to the shape can be generated until a steady solution is obtained:
240
(1) Evolution step: Form the combination C(x,y)
= A(x,y) - ~t(2~t)-!aB(x,y)
where A(x, y) is as in (11) and B(x, y) is as in (13). (2) Fidelity step: Set
{C(X'Y)
~ (2~t)! (v'e -
Jo)} U (EO n (D \ D))
An example illustrating the algorithm appears in Figure 4. The algorithm effectively imposes Dirichlet boundary conditions on aD by forcing the solution to coincide with the known image information outside of 13 at the end of each time step. This is a convenient but not necessarily the most accurate treatment of boundaries; indeed, when Ilt is taken to be larger, as shown in the third plot of Figure 4, kinks in the solution may be introduced.
°0
0,
0-2
OJ
O'
05
06
0.7
08
09
,
Fig. 4. Image reconstruction example. The central square is the damaged region in the left image. Taking the initial region r;o to be the union of the missing region and the two rectangular subregions and a = 1 leads to the reconstructed bent bar shown in the second image from the left (time t = 0.05); this result was very carefully computed with time step-size 0.00001/1024 using 128 x 128 basis functions. The rightmost was obtained by repeating the calculation with very large time steps, and demonstrates that the algorithm remains stable (albeit, of course, at the expense of accuracy).
3.5. Junctions Similar to the original threshold dynamics of Merriman, Bence and Osher [15,16), the algorithm for Willmore flow extends naturally to multiple junctions even though the asymptotic expansions require smoothness of the interfaces. This extension involves convolving the characteristic function for
241
each region with the appropriate kernel, and thresholding according to the maximum point-wise value of the convolutions. More precisely, let Ei C lR 2 denote the set of the ith region (2 :s; i :s; N) whose boundary is to be moved via Willmore flow and set E~ := E i . We then alternate the following steps for k = 0,1,2, ... in order to generate the approximations Ef to the flow at the discrete times tk = k . L:.t: (1) Form the convolution: for 1 :s; i :s; N, Ai(x, y)
lEf(x,y)
=
* (2L:.t)t (VOG~/(/X,Y) - JoG6~(X'Y))'
where Gt(x, y) is the fundamental solution of the heat equation:
(2) Set:
Ef+l = {(x, y)
E
lR 2
:
Ai(x, y) ~ Aj(x, y), i
# j} ,1 :s; i :s; N
An illustration of junctions evolving by Willmore flow is given in Figure 5. The initial system is taken to be 4 regions meeting at a point. The symmetry of this configuration suggests that it is an unstable equilibrium: different nearby initial data would be expected to diverge. Our numerical algorithm perturbs this state into a lower energy configuration, and leads to the evolution shown. Note that our simulation makes use of homogeneous Neumann boundary conditions; this explains why curves meet the boundary at right angles in the final configuration. We are not aware of any analytical work concerning Willmore flow of junctions.
°0
III
0:1
03
o.
01
O'
01
01
01
Fig. 5. Evolution according to Willmore flow. Unstable junction at time t = 0 (left) breaks into 2 symmetric triple junctions by time t = 0.01 (right). The time step-size was taken to be L:.t = 0.01/10000.
242
3.6. General Curvature Motion Using Signed Distance Functions and Redistancing In this section, we present some examples diffusing signed distance functions and redistancing. We shall see that the resulting algorithm is accurate, robust and flexible in the sense that a wide range of diffusion times. In the following numerical study, we take f(",) = ",1/3. A circle shrinks from its initial radius TO according to
T(t) =
4/3
(
TO
4)
-:it i
Using the same setting as the above, we obtain Table 3. Next, we compute Table 3. Numerical convergence study of interface motion by cubic root of curvature. 2
4
8
16
32
7.631e-4
3.462e-4
1.619e-4
7. 563e-5
3.365e-5
s=
Abs. err
1. 587e-3
flows of a curve (trefle shape) that contains inflection points. We evolve a polar curve defined by 1
r(O) = 4(1
1
+ 2 sin(30)).
Figure 6 shows two such results.
120
60
100
50
80
40
60
30
40
20
20
10
20
40
60
Fig. 6.
80
100
120
Motions of a trefie by
10
,,1/3
20
30
and
1"1 1 / 3 .
40
50
60
243 4. Conclusions In this paper threshold dynamics algorithms for the high-order geometric motion of curves are investigated. Our methods have several characteristic properties: (a) a very simple structure which consists of alternating convolution and thresholding steps; (b) excellent stability properties in the use of large time steps, and in practice did not exhibit any apparent instabilities. Mathematically, this article as well as earlier studies [23-28,31J hint at the diversity of motions possible under threshold dynamics. Rigorous mathematical studies of such motions would be of great interest. Given the excellent stability properties exhibited by the algorithms, and the diversity of motion laws that are possible, we consider this class of algorithms to be interesting candidates for the treatment of certain geometric motion laws arising in image processing and other related fields.
References 1. T. J. Willmore, Riemannian Geometry (Clarendon Press, 1993). 2. M. Kass, A. Witkin and D. Terzopoulos, International Journal of Computer Vision 1, 321 (1987). 3. S. Masnou, IEEE Trans. Image Processing 11, 68 (2002). 4. S. Masnou and J.-M. Morel, Level lines based disocclusion, in 5th IEEE International Conference on Image Processing, 1998. 5. M. Nitzberg, D. Mumford and T. Shiota, Filtering, segmentation, and depth, in Lecture Notes in Computer Science, (Springer-Verlag, 1993) 6. S. Esedoglu and R. March, Journal of Mathematical Imaging and Vision 18, 7 (2003). 7. M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Siggraph 2000, Computer Graphics Proceedings, ed. K. Akeley (ACM Press I ACM SIGGRAPH I Addison Wesley Longman, 2000). 8. E. Bansch, P. Morin and R. Nochetto, International meeting on free boundary problems, theory and applications, Trento (2002). 9. E. Dogan, P. Morin, R. Nochetto and M. Verani, Preprint (2006). 10. M. Droske and M. Rumpf, Interfaces Free Bound. 6, 361 (2004). 11. M. Burger, J. Comput. Phys. 203, 602 (2005). 12. S. Esedoglu and J. Shen, European J. Appl. Math. 13, 353 (2002). 13. K. Deckelnick, G. Dziuk and C. Elliott, Acta Numerica 14, 139 (2005). 14. K. Deckelnick and G. Dziuk, Interfaces and Free Boundaries 8, 21 (2006). 15. B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, in Computational Crystal Growers Workshop, ed. J. Taylor (American Mathematical Society, Providence, Rhode Island, 1992) pp. 73-83. Also available as UCLA CAM Report 92-18, April 1992. 16. B. Merriman, J. Bence and S. Osher, J. Comput. Phys. 112, 334 (1994). 17. S. Esedoglu, S. Ruuth and R. Tsai, Threshold dynamics for shape reconstruc-
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18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
32. 33. 34. 35.
36.
tion and disocclusion, in Proc. ICIP05, International Conference on Image Processing, (Genova, Italy, 2005). S. Esedoglu, S. Ruuth and Y. Tsai, Interfaces and Free Boundaries 10 (2008). L.-T. Cheng and Y.-H. Tsai, J. Comput. Phys. 221, 4002 (2008). S. J. Ruuth, J. Comput. Phys. 144,603 (1998). L. Evans, Indiana University Mathematics Journal 42, 553 (1993). G. Barles and C. Georgelin, SIAM Journal on Numerical Analysis 32, 484 (1995). P. Mascarenhas, Diffusion generated motion by mean curvature, CAM Report 92-33, University of California, Dept. of Math (Los Angeles, 1992). S. J. Ruuth, J. Comput. Phys. 145, 166 (1998). H. Ishii, G. Pires and P. Souganidis, J. Math. Soc. Japan 51, 267 (1999). S. J. Ruuth and B. Merriman, SIAM Journal on Applied Mathematics 60, 868 (2000). S. J. Ruuth, B. Merriman and S. Osher, J. Comput. Phys. 151, 836 (1999). S. J. Ruuth and B. Wetton, J. Scientific Computation 19, 373 (2003). B. Jawerth and P. Lin, J. of Visual Communication and Image Representation 13, 94 (2002). S. Esedoglu and Y. Tsai, J. Comput. Phys. 211, 367 (2006). R. Grzibovskis and A. Heintz, A convolution-thresholding scheme for the Willmore flow, preprint 34 (revised version) (Chalmers Univ. of Tech, Goteborg, Sweden, 2003). T. F. Chan, S. H. Kang and J. Shen, J. Appl. Math. 63, 564 (2002). S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer-Verlag, New York, 2002). S. Esedoglu, S. Ruuth and Y. Tsai, Preprint . H. Ishii, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, in Curvature flows and related topics (Levico, 1994), GAKUTO Internat. Ser. Math. Sci. Appl. Vol. 5 (Gakk6tosho, Tokyo, 1995) pp. 111-127. P. Smereka, J. Sci. Comput. 19, 439 (2003).
245
BLOW-UP PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS OF PARABOLIC TYPE· Yujuan Chen
Department of Mathematics, Southeast University, Nanjing 210018, P. R. China and Department of Mathematics, Nantong University, Nantong 226007, P. R. China • E-mail: [email protected] Mingxin Wang
Department of Mathematics, Southeast University, Nanjing 210018, P. R. China and School of Mathematical Science, Xuzhou Normal University, Xuzhou 221116, P. R. China • E-mail: [email protected] In this survey paper, we shall give a brief account on some aspects of blow-up problems for partial differential equations and systems of parabolic type with nonlinear boundary conditions, localized terms, nonlocal terms and problems not in divergence form. We are mainly concerned with blow-up rates, blowup sets, blow-up profiles, blow-up boundary layers, total blow-up, single point blow-up and simultaneous blow-up.
Keywords: Parabolic Equations and Systems; Initial and Boundary Value Problems; Blow-up Problems.
*This work was supported by PRC Grants NSFC 10771032, and the Natural Science Foundation of Jiangsu province BK2006088, Jiangsu Education Commission 07KJDl10166, and the postdoctoral project of Jiangsu province 0702004C.
246 1. Introduction
We first consider some examples of ordinary differential equations:
y'(t) = y(t), t > 0; y'(t) = y2(t), t > 0; y'(t)=ey(t), t > 0; y'(t) = y(t) - y2(t), t > 0;
yo> 0,
(1)
y(O) = Yo> 0, y(O) = Yo > 0, y(O) = Yo> O.
(2)
y(O)
=
(3)
(4)
• For the problem (1), the solution is y(t) = yoe t , which is defined in [0, 00) and called a global solution. Clearly limt ..... oo y(t) = 00. • For the problem (2), the solution is y(t) = (T - t)-l, where T = Yolo This solution is defined in [0, T) U (T, 00) and satisfies limt .....T y(t) = 00; we say that the solution y(t) blows up in finite time T. Moreover, y(t) --. 00 with the rate (T - t)-l when t --. T. Such rate (T - t)-l is called the blow-up rate. • For the problem (3), the solution is y(t) = In T~t' where T = e- Yo • This solution is defined in [0, T), and blows up in finite time T and the blow-up rate is In T~t. • For the problem (4), y(t) = et [(l - YO)Yol + etl-l. Clearly limt ..... oo y( t) = l. Ordinary differential equations is the simplest evolution ones. Similar phenomena also occur in partial differential evolution equations, and topics such as global solutions, blow-up in finite time, blow-up rate and blow-up set have been extensively studied for such equations. In this survey paper, we shall give a brief account on some aspects of these topics for partial differential equations and systems of parabolic type. Since the role of critical exponents in blow-up theorems have already been discussed by Deng and Levine 23 ,46, we only focus on the following problems in this paper: • Blow-up rates, blow-up sets, blow-up profiles and blow-up boundary layers; • Simultaneous blow-up; • Systems with nonlinear boundary conditions; • Problems with localized terms, nonlocal terms and not in divergence form; • Total blow-up and single point blow-up. Moreover, we will be mainly concerned with some recent progresses on these topics that are related to our own work. Well known results such as those obtained by Chadam et a18 , Friedman and McLeod32 , Giga and Kohn 36 - 38 , and Hu 42 will not be considered here, though many of the ideas
247
and techniques in the recent progresses discussed here originate from these early works. Throughout this paper, unless otherwise specified, n denotes a bounded domain in jRN with smooth boundary an, r] is the unit outward normal vector on an. 2. Simultaneous blow-up In the study of blow-up rates, blow-up sets, blow-up profiles and blow-up boundary layers of blow-up solutions for systems, very often components of the solution blow up simultaneously. This phenomenon is called simultaneous blow-up. So, we first survey some results on this aspect. • Semilinear systems Souplet and Tayachi 70 studied the initial value problem to the following system
(5) with m, n, p, q > 1. They gave the conditions for simultaneous and nonsimultaneous blow-up: (i) If m > q + 1 or n > p + 1, then there exist initial data Uo and vo such that non-simultaneous blow-up occurs; (ii) If m < q + 1 and n < p + 1, then simultaneous blow-up occurs for every initial data. The critical cases m = q+ 1, n = p+ 1 was not considered. For radially symmetric and nondecreasing in time solutions, they obtained two kinds of simultaneous blow-up rates. Rossi and Souplet67 studied the system (5) in a bounded domain with homogeneous Dirichlet boundary conditions. It was found that in the region m > q + 1 and n > p + 1 (where both u and v can blow up alone), there exist initial data such that simultaneous blow-up occurs. • Semilinear systems with nonlinear boundary conditions Recently, Li et alSO studied the simultaneous and non-simultaneous blow-up for heat systems with coupled nonlinear boundary fluxes: Ut
= Au,
au 8 =um+v p ,
{
u~, 0) = uo(x),
Vt
= Av,
OV
Or]
= u q + vn ,
v(x,O)
En x (O,T), (x, t) E an x (0, T),
(x,t)
= vo(x), x
E
n,
where parameters m, n ~ 0, p, q > 0, initial data uo(x) and vo(x) are positive smooth functions and satisfy the compatibility conditions ~ =
248
Un + vb, ~ = u6 + v(f on an. Their main results for the simultaneous and non-simultaneous blow-up can be summarized in the following two figures: n=~
n
n=~
n N2
p+
C1
N2
85
83
p+
C2
83
q p+1 q+1
E1
m=P+f=n
E1 q(p+1 q+1
81 G
lliill 1 1 v+
Case (a)
* *
82
82
N1
q+l
pq:::; 1
m
lliill v+1
Case (b)
N1
q+l
m
pq> 1
A necessary and sufficient condition for global existence: closed region G. For radially symmetric and nondecreasing in time solutions, a necessary and sufficient condition for the existence of non-simultaneous blow-up: regions N 1 ,N 2, C 1 -C 3 ,El,E2. Conditions which guarantee that blow-up is non-simultaneous for every initial data: N 1: U blows up while v remains bounded. N 2 : v blows up while u remains bounded. Conditions under which both simultaneous and non-simultaneous blow-up are possible: C 1 : u may blow up alone for large Uo; v may blow up alone for large Vo; There exist initial data such that simultaneous blow-up occurs. C 2 : There exist initial data such that simultaneous blow-up occurs; u may blow up alone for large Uo. C 3 : There exist initial data such that simultaneous blow-up occurs; v may blow up alone for large Vo. Non-simultaneous blow-up occurs under suitable initial data: E 1 : u blows up alone. E 2 : v blows up alone. Conditions which guarantee simultaneous blow-up for every initial
*
*
* *
249
data: regions 8 1-8 6 , Immediately after, Zheng et al 97 considered the following cross-coupled parabolic system Ut
= ~u + u m + vP , Vt =
~v,
au _ O'T] - 0,
O'T] - u
u(x,O) = uo(x),
v(x,O) = vo(x), x
{
ov _ q + v,k
(x, t) En x (0, T), (x, t) E E
an x (0, T),
n,
where m, k 2: 0, p, q > 0, initial data uo(x) and vo(x) are positive smooth functions and satisfy the compatible conditions. Their main results concerning the simultaneous and non-simultaneous blow-up are the following: 1° If p + 2 < 2k, then for given uo, there exists large Vo such that v blows up while u remains bounded. Conversely, if v blows up and u remains bounded, then p + 2 < 2k. 2° If u blows up and v remains bounded, then 2q + 1 < m. Conversely, if 2q + 1 < m with n = 1, then for given vo, there exists large Uo such that u blows up while v remains bounded. 3° If m < 2q + 1 and 2k < p + 2, then simultaneous blow-up occurs for all positive initial data. 4° If m ::; 1 and p + 2 < 2k, then v blows up and u remains bounded for every positive initial data. If k ::; 1, 2q + 1 < m, and n = 1, then u blows up and v remains bounded for every positive initial data. 5° If p + 2 < 2k, 2q + 1 < m and n = 1, then there may occur both simultaneous and non-simultaneous blow-up. • 8emiIinear systems with nonlinear source terms and nonlinear boundary conditions Recently, Li et a1 51 considered the following multi-coupled parabolic system Ut Vt
= ~u + v P + u m , = ~v + u q + v n ,
au I ov k O'T] = v, O'T] = u ,
(x, t) E B x (0, T), (x, t) E B x (0, T),
(x, t)
E oB x (0, T),
(6)
u(x, 0) = uo(x), v(x,O) = vo(x), x E B, where B = {Ixl < I}. They obtained the following results for the simultaneous blow-up problem. 1° There exist initial data such that non-simultaneous blow-up occurs if and only if m > max{q, 2k} + 1, or n > max{p, 2l} + 1.
250
2° Non-simultaneous blow-up occurs for every initial data if either m > max{q,2k} + 1 and n ~ 1, or n > max{p,21} + 1 and m ~ 1. 3° Both non-simultaneous and simultaneous blow-up may occur if m > max{ q, 2k} + 1 and n > max{p, 21} + 1. • Semilinear systems with localized nonlinear terms Zheng et al g8 studied the following problem with localized terms
Ut = L\u + um(x, t) + vP(O, t), Vt = L\v + uq(O, t) + vn(x, t),
(x, t) E B x (0, T), (x, t) E B x (0, T),
u = v = 0,
(x, t) E aB x (0, T),
!
u(x,O) = uo(x),
v(x,O) = vo(x), x
E
(7)
B,
where constants m, n,p, q > 1, initial data uo, Vo E C 2 (Q) n C(O) are nontrivial and nonnegative functions satisfying the compatibility conditions. They obtained the following results on the simultaneous blow-up: 1° If m < q + 1 and n < p + 1, then any blow-up solution must be simultaneous blow-up. 2° If m > q + 1, then there exist initial data such that u blows up while v remains bounded. Conversely, if u blows up and v remains bounded, then m>q+1. 3° If n > p + 1, then there exist initial data such that v blows up while u remains bounded. Conversely, if v blows up and u remains bounded, then n>p+1. 4° If m > q + 1 and n > p + 1, then both simultaneous and nonsimultaneous blow-up may occur. Some related works on simultaneous blow-up, please refer to papers of Li and Wang 52- 54 . 3. Systems with homogeneous Dirichlet boundary conditions Consider the semilinear problem
Ut = L\u + uQv P, Vt = L\v + u qv f3 ,
! u
= v = 0,
x E Q, t > 0,
x E Q, t > 0, aQ, t > 0,
x E
(8)
u(x, 0) 2 0, u(x, 0) 20, XEQ,
where parameters a, j3, p and q are nonnegative, and satisfy a + p > 0, j3 + > 0. The initial values uo(x) and vo(x) are continuous and satisfy the compatibility conditions. q
251
When 0 = ]Rn, Escobedo and Levine 28 ,47 obtained the critical Fujita exponents for the Cauchy problem of (8). When a = (3 = 0, Escobedo and Herrero 26 ,27 studied the problem (8) and the corresponding Cauchy problem (0 = ]Rn), and the critical exponents were obtained. Fila and Quittner30 studied the Cauchy problem of a system with k components and gave the blow-up rate estimate of blow-up solution. Chen lO discussed problem (8) for the special case: p, q > O,p > (3 - 1 and q > a-I, and obtained that if
a:::; 1,(3:::; 1 and pq:::; (1- a)(l - (3),
(9)
then all solutions of (8) exist globally, and if (9) does not hold, then the solution of (8) exists globally for small initial data, while the solution of (8) blows up in finite time for large initial data. Later, Wang 76 studied the problem (8) for the general case. Let Al be the first eigenvalue of -~ in 0 with homogeneous Dirichlet boundary condition, and ¢(x) the corresponding eigenfunction. Then Al > 0, ¢(x) > in 0 and ~~ < on 80, here T/ is the unit outward normal vector. Wang 76
°
°
obtained the following results: (i) Assume that
a:::; 1,(3 :::; 1 and
pq:::; (1 - a)(l - (3).
(10)
Then all solutions of (8) exist globally. (ii) Assume that
a> 1,p > O,q = 0,(3 = 1,A1 < 1,a:::; 1 + p(l- Ad/AI,
(11)
or (3
> 1,q > O,p = O,a = 1,A1 <
1,(3:::; 1 +q(l- A1)/A1.
(12)
Furthermore, if a = 1 +p(l- Ad/ Al in (11) or (3 = 1 +q(l- A1)/ Al in (12), we also assume Al < 2/3. Then, for any initial data uo(x) 2: 0, =j. 0, vo(x) 2: 0, =j. 0, the solution of (8) blows up in finite time. (iii) If (10), (11) and (12) do not hold, then the solution of (8) exists globally for small initial data, and blows up in finite time for large initial data. Open problem If 2/3:::; Al < 1, and a = 1 + p(l - Al)/Al in (11) or (3 = 1 + q(l - Ad/AI in (12), we do not know whether or not the solution of (8) blows up in finite time for every uo(x) 2: 0, =j. 0, vo(x) 2: 0, =j. 0. Assume that the solution (u, v) of (8) blows up in finite time T. For the blow-up rate of (u, v), the results of Caristi and Mitidietri7, Deng 20 ,
252
Wang 77 ,78, and Zheng96 are summarized as: c
~
max
m~u(.,
r)(T -
t)(Hp-,B)/[pq-(l-a)(l-,B)] ~
C,
r)(T -
t)(Hq-a)/[pq-(l-a)(l-{3)] ~
C
O~r~t xEO
C ~
max
m~v(·,
O~r~t xEO
for some positive constants c and C. 4. Nonlinear Boundary Value Problem • Single equation with nonlinear source The simplest single equation with nonlinear boundary condition is the following Ut
!
=
6.u,
x E
au _ a -u, p
u~, 0) = UO(X) ;::: 0,
n, t > 0,
x EOn, t > 0,
(13)
with p > 0. The pointer works on the global existence and finite time blow-up of solutions to the problem (13), to our knowledge, are the works of Levine and Payne 48 , and Walter 71 • It was proved that every positive solution exists globally if and only if p > 1. Later, Gomez et al 39 (for the case n = BR(O)), and Hu and Yin 43 proved that the blow-up occurs only on the boundary an. Chlebik and Filal 4, Deng 17 , and Fila and Quittner29 studied the blow-up rate of the blow-up solution to the general form of the problem (13). Hu 41 studied the non-degeneracy and single point blow-up for solution of the problem (13). For the following single equation with nonlinear source term and nonlinear boundary condition:
+ uP, ux(O, t) = 0, u x (l,
Ut
=
U
°<
xx
{
u(x, 0) = uo(x) ;::: 0,
t)
= u q (l, t),
X
< 1, t > 0,
t > 0,
°
(14)
~ x ~ 1,
where the parameters p and q satisfy max{p, q} > 1. By the results of Wang 90 it is known that all positive solutions blow up in finite time. An interesting problem is to consider the influences of the source term uP and the nonlinear boundary term u q on the blow-up rate estimates. Let (u, v) be the blow-up solution of (14) and T be the blow-up time, Lin and Wang 57
253 proved that c:S supu(x, t)(T - t)a :S C for some positive constants c and C, where a = 1j(p - 1) if p 2: 2q - 1, while a = 1j[2(q - l)J if P < 2q - 1. Moreover, it was also proved that blow-up occurs only at the boundary x = 1. The problem Ut
= U xx
0
).uP,
-
< x < 1, t > 0,
ux(O, t) = 0, u x (1, t) = uq(l, t), t > 0, {
°
u(x, 0) = uo(x) > 0,
:S x :S 1
°
66
was studied by Rossi , where p, q > 1. For the case < ). < q, it was shown that u(l, t) rv (T - t)-1/(2(q-l» if p :S 2q - 1. For the case). < 0, it was shown that u(l, t) rv (T - t)-l/(p-l) if p 2: 2q - 1. • Systems without source terms Consider the following problem
: ~u',
x E
n, t > 0,
xE
an,
u(x, 0) = uo(x) 2: 0,
x E
n,
Ut -
6.u
:-~v,:
!
Vt
= 6.v,
t > 0,
(15)
where parameters p and q satisfy pq > 1. Wang and Wang 86 proved that the solution blows up in finite time. Deng 20 obtained the following blow-up rate estimates: c:s max maxu(·, T)(T - t)(l+p)/[2(pq-l)] :S C, 0:,>1"9 BR
c:s max maxv(·, T)(T -
t)(l+q)/[2(pq-l)]
:S C.
o:'>1":,>t BR
• Systems with source terms Consider the following systems with nonlinear source terms and nonlinear boundary conditions: Ut
= U xx + v P ,
°< x < °< <
1, t > 0,
= Vxx + uk, x 1, t > 0, Q ux(O, t) = 0, u x (1, t) = v (l, t), t > 0,
Vt
vx(O, t) = 0, vx (1, t) = u m(l, t), t > 0, u(x, 0)
= uo(x),
v(x, 0)
= vo(x),
O:S x :S 1,
(16)
254
where p, q, k and m are positive constants, initial data uo(x) and vo(x) are nonnegative and nontrivial C 1 functions and satisfy the following conditions
UOx(O) = 0, uo x (l) = vZ(l), vOx(O) = 0, vo x (l) = ug'(l), u~(x) 2: 0, u~(x) + vb 2: 0, v~(x) 2: 0, v~(x) + u~ 2: 0. It follows from the classical results that the solution (u, v) of (16) satisfies
Ut(x, t), Vt(x, t), ux(x, t), vx(x, t) 2: 0. So, maxu(·,t) ~,~
= u(l,t)
and maxv(·,t) ~,~
= v(l,t).
Wang et algO proved that
the solution (u, v) of the problem (16) blows up in finite time if and only if
max{pk, mp, mq, qk} > l. For the blow-up rates and blow-up sets of blow-up solutions to the problem (16), Wang82 obtained the following results: 10 There exist positive constants c and C such that the following hold. (i) When p 2: (2qk + 2q - 1)/(2 + k) and k 2: (2mp + 2m - 1)/(2 + p), then c ::; u(l, t)(T - t)(1+p)/(pk-l) ::; C, {
c::; v(l,
t)(T - t)(l+k)/(pk-l) ::;
C;
(ii) When p 2: (2qk + 2q - 1)/(2 + k) and k < (2mp + 2m -1)/(2 + p). If p 2: (2mq + q - 1)/(1 + m), then
c::; u(l, { c ::; v(l, If p
t)(T - t)(2+ p )/[2(m p-l)] ::;
C,
t)(T - t)(1+2m)/[2(m p-l)] ::;
C;
< (2mq + q - 1)/(1 + m), then c ::; u(l, t)(T - t)(1+q)/[2(m q-l)] ::; C, { c ::; v(l, t)(T - t)(l+m)/[2(m q-l)] ::; C;
(17)
(iii) When p < (2qk + 2q -1)/(2 + k) and k 2: (2mp+ 2m -1)/(2 + p). If k 2: (2mq + m - 1)/(1 + q), then
c::; u(l, { c::; v(l, If k < (2mq
t)(T - t)(1+ 2q)/[2(qk-l)] ::; t)(T - t)(2+k)/[2(qk-l)] ::;
+ m -1)/(1 + q),
then (17) holds.
C,
C;
255
(iv) When p < (2qk + 2q -1)/(2 + k) and k < (2mp + 2m -1)/(2 + p), (17) holds. 2° If the inequalities p ~ (2qk + 2q - 1)/(2 + k) and k ~ (2mp + 2m1)/(2 + p) do not hold simultaneously, then the blow-up set consists of a single point x = l.
• Quasilinear systems without source terms Consider the following quasilinear systems with nonlinear boundary conditions: x E
n,
x
an,
E
x E
t > 0, t > 0,
(18)
n,
where the indices mi are positive, and mij are non-negative, i,j = 1"" ,k. In addition, initial data UiQ (x) (1 ::; i ::; n) are positive C1 functions and satisfy the compatibility conditions. When k = 1, Wang 75 proved that: If m1 ::; 1, then all positive solutions of (18) exist globally if and only if mll ::; m1; if ml > 1, then all positive solutions of (18) exist globally if and only if 2mll ::; m1 + l. When k = 2, C. V. Pao61 established some sufficient conditions on the existence of global solutions and finite time blow-up by constructing upper and lower solutions. When ml = m2 = 1, Wang 74 showed that all positive solutions of (18) exist globally if and only if mll ::; 1, mn ::; 1 and m12m21 ::; (1 - mll)(1 - m22), and Rossi 65 studied the blow-up rtate of the blow-up solution. Later, Wang et a1 85 studied problem (18) for the slow diffusion case: ml ::; 1, m2 ::; 1, and proved that all solutions of (18) exist globally if and only if mll ::; m1, m22 ::; m2 and m12m21 ::; (ml mll)(m2 - mn). Wang80 studied the problem (18) for the fast diffusion case: m1 ~ 1, m2 ~ 1, and the fast-slow diffusion case: m1 ~ 1, m2 ::; 1, and derived certain sufficient conditions for the global existence of all solutions. The conditions obtained by him are also necessary if n = B(O, R) is the ball in ]RN centered at the origin with radius R. When k > 2, the problem (18) was discussed by Wang et a1 86 for the special case of mi = 1, 1 ::; i ::; k. Denote M = (mij) k x k, it was proved that all positive solutions of (18) exist globally if and only if all principal minor determinants of 1- M are non-negative. Later, Wang et a1 90 ,91 studied the problem (18) for the slow diffusion case: mi ::; 1, 1 ::; i ::; n, and deduced
256
the necessary and sufficient conditions for the global existence of all positive solutions. Recently, Qi et a1 64 studied the more general case, i.e., the fast diffusion case: mi ?: 1 for all i, and fast-slow diffusion case: mi ?: 1 and mj ~ 1 for some i,j. Denote M = (mij)kxk, bi = min {mi' (mi + 1)/2}, bij = bi 8ij , i,j = 1"" ,k, and set B = (bij)kxk, F = B-M. They obtained the following results: 1° If all the principal minor determinants of F are non-negative, then all solutions of (18) exist globally. 2° If one of the following three conditions holds, then all solutions of (18) blow up in finite time: (i) There exists i : 1 ~ i ~ n such that bi < miij (ii) F has a negative principal minor determinant in which bi ~ Ij (iii) There exists a negative principal minor determinant of F and n = B(O, R) is the ball in ]RN centered at the origin with radius R. Remark: For the special case: n = B(O, R), these two theorems show that all solutions of (18) exist globally if and only if all the principal minor determinants of F are non-negative. For the blow-up rate estimates of systems with more than two components and nonlinear boundary conditions, the reader can refer to the works of Wang 78 ,79,83 .
• Systems in half space Upper bounds of blow-up rates for blow-up solutions to the following problem XE]R~, Xl
t>O,
= 0, t > 0,
(19)
u(X, 0) = uo(x)?: 0, v(x,O) = vo(x) ?: 0, X E ]R~, uo, Vo E 2:,"0 (]R~)
was discussed by Chlebik and Fila 13, where constants 8i E {O, I}. Let T be the blow-up time of solution (u, v) and C be a suitable positive constant. Their results read as follows: 1° When 81 = 82 = and 83 = 84 = 1. If mq > 1 and max{1 + m, 1 + q} /(mq - 1) ?: N, then
°
u(X, t)
~ C(T_t)-(1+q)/[2(m Q-1)J,
v(x, t)
~ C(T_t)-(1+ m)/[2(m Q-1)J,
2° When 81 = 84 = 1 and 82 = 83 = 0. If mp > 1 and if moreover, one of the following holds:
0< t < T
U X1 ,VX1 ~
0, and
257
(i) max{p + 2, 2m + l}/(mp - 1) > N, (ii) max{p + 2, 2m + l}/(mp -1) = Nand m, p ~ 1, then u(x, t) ~ C(T_t)-(P+2)/[2(mp-1)J, vex, t) ~ C(T_t)-(2m+1)/[2(m p-1)J, 0 < t < T.
For the case 81 = 82 = 0, 83 = 84 = 1, under the condition mq > 1, Deng et a1 21 proved that if max( a,,8) ~ N /2, all nontrivial nonnegative solutions are nonglobal; whereas if max(a,,8) < N /2, then there exist both global and nonglobal nonnegative solutions, where a = ~(q + l)/(mq - 1) and ,8 = ~ (m + 1) / (mq - 1). When N = 1, they established some results for the blow-up rate for the non-global solutions and some results for the decay rate for the global solutions (in the supercritical case). They also constructed a nontrivial solution with vanishing initial values when mq < 1. Wang et a1 93 , for the case N = 1, obtained lower bounds of blow-up rates of solutions to the problem (19), and improved the results of Deng 21 • The general case of the problem (19), i.e. 81 = 82 = 83 = 84 = 1, was considered by Li and Wang 55 . Let the solution (u,v) of the problem (19) blow up in finite time T, they obtained the following results: 10 There exists a constant c > 0 such that the following hold. (i) When p ~ (2qk + 2q - 1)/(2 + k) and k ~ (2mp + 2m - 1)/(2 + p), then max sup u(x, r) ~ c(T - t)-(l+p)/(pk-1), O::;r::;t IR~
{ max supv(x, r) ~ c(T - t)-(l+k)/(pk-1), O::;r::;t IR~
0< t < T, 0< t < T.
(ii) When p ~ (2qk + 2q - 1)/(2 + k) and k < (2mp + 2m - 1)/(2 + p). If p ~ (2mq + q - 1)/(1 + m), then max supu(x, r) ~ c(T - t)-(2+p)/[2(m p-1)J, O::;T::;t IRN
{ max s;pv(x, r) O::;T::;t IR~
~ c(T -
t)-(1+2m)/[2(mp -1)J,
0< t < T, 0< t < T;
if p < (2mq + q - 1)/(1 + m), then max supu(x, r) ~ c(T - t)-(l+q)/[2(m q-1)J, O::;r::;t IR~
{ max supv(x, r) ~ c(T O::;T::;t IR~
(iii) When p < (2qk
+ 2q -
t)-(l+m)/[2(m q -l)J,
0< t < T, 0< t < T.
(20)
1)/(2 + k) and k ~ (2mp + 2m - 1)/(2 + p).
258
If k ;::: (2mq + m - 1)/(1 + q), then
max sup u(x, T) ;::: c(T - t)-(1+2q)/[2(qk-l)] , 0< t < T, lR!;' max sup v(x, T) ;::: c(T - t)-(2+k)/[2(qk-l)], 0< t < T; { O::;r::;t lR-;: O::;r::;t
if k < (2mq + m - 1)/(1 + q), then (20) holds. (iv) When p < (2qk + 2q -1)/(2 + k) and k < (2mp + 2m - 1)/(2 + p), (20) holds. 2° There exists a constant C > 0 such that the following hold. (i) Assume that p;::: (2qk+2q-1)/(2+k) and k ;::: (2mp+2m-1)/(2+p). If max{l + p, 1 + k}/(pk -1) ;::: N/2, then max sUpU(X,T):S C(T - t)-(l+p)/(pk-l), lR!;' { max supv(x, T) :S C(T - t)-(l+k)/(pk-l), O::;r::;t lR!;'
0 < t < T,
O::;r::;t
0 < t < T.
(ii) Assume that p ;::: (2qk+2q-1)/(2+k) and k < (2mp+2m-1)/(2+p). For the case p;::: (2mq+q-1)/(1+m), ifmax{p+2, 1+2m}/(mp-1) > N, or max{p + 2,1 + 2m}/(mp - 1) = Nand m, p;::: 1, then
max supu(x, T) :S C(T lRN { max su+pv(x, T) :S C(T O::;r::;t lR-;:
t)-(2+p)/[2(m p -l)],
O::;r::;t
t)-(1+2m)/[2(mp -l)],
0< t < T, 0< t < T;
For the case p < (2mq + q - 1) / (1 + m), if max {I + q, 1 + m} / (mq - 1) ;::: N, then max supu(x, T) :S C(T - t)-(l+q)/[2(m q -l)], 0 < t < T, IR-;: q { max sUpV(X,T):S C(T_t)-(l+m)/[2(m -l)], 0 < t < T. O::;r::;t IR!;' O::;r::;t
(21)
(iii) Assume that p < (2qk+2q-1)/(2+k) and k ;::: (2mp+2m-1)/(2+ p). For the case k ;::: (2mq+m-1)/(1+q), ifmax{1+2q, 2+k }/(qk-1) > N, or max{l + 2q, 2 + k}/(qk -1) = Nand q, k;::: 1, then max supu(x, T) :S C(T - t)-(1+2 q)/[2(qk-1)], 0 < t < T, lR-;: max supv(x T) < C(T - t)-(2+k)/[2(qk-l)] 0 < t < T· { O::;r::;t IRN ' " O::;r::;t
+
For the case k < (2mq+m-1)/(1 +q), if max{l +q, 1 +m}/(mq-1) ;::: N, then (21) holds.
259
(iv) Assume that p < (2qk + 2q - 1)/(2 + k) and k < (2mp + 2m 1)/(2 + p). If max{1 + q, 1 + m}/(mq -1) ~ N, then (21) holds. 3° For the cases (ii), (iii) and (iv) of 1°, blow-up occurs only on the boundary More precisely, if Do C is such that no c then
alR:t.
lR:t
lR:t,
sup {llu(., t)lIc(fl o) + Ilv(·, t)llc(flo)} <
00.
09
Wang et a1 94 using the self-similar solution to construct the upper and lower solutions, studied the critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition in the half line. Zhou et al lOO using the upper and lower solutions method discussed the critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. Huang et al 44 studied the critical Fujita exponents for the porous medium equation with a nonlinear boundary condition. 5. Initial and boundary value problems with localized terms • Single equation The blow-up properties of solutions to the single equation
Ut - D..u = f(u(xo(t), t)), xED, {
u(x, t) = 0, u(x, 0)
=
x
uo(x)
~
0,
E
t > 0,
aD, t > 0,
(22)
xED
have been discussed by many authors, see for example, Cannon and Yin 6 , Chadam et al8 , Souplet68 ,69, and Wang and Chen72 , here xo(t) is a given function of t. Souplet68 obtained a sharp critical blow-up exponent for the problem (22). Souplet 69 introduced a new method to investigate the profile of the blow-up solution and estimates on the size of the boundary layer and on the asymptotic behavior of the blow-up solution in the boundary layer. First for the blow-up rate, he proved that if f(u) = uP with p > 1 then lim (T - t)l/(P-l)U(X, t) t-+T
= lim (T -
t)l/(p-l)
Ilu(t)ll= = (p - 1) -l/(p-l)
t-+T
uniformly on the compact subset of D, and if f(u) = eU then lim I1n(T - t)I-1u(x, t)
t-+T
= t-+T lim Iln(T -
t)I- 1Ilu(t)lI=
=1
uniformly on the compact subset of D, where T is the blow-up time of u.
260
For the case feu) = uP with p > 1, he also obtained the estimate on the size of the boundary layer: C 1 Ji=h"U(t)lloo
~ u(x, t) ~ C2 Ji=h"U(t)lloo
for all (x, t) E n x [to, T) such that d(x) ~ K VT - t, where the constants C 2 2:: C 1 > and K > 0. For the boundary layer estimates, if feu) = uP with 1 < p < 2, he obtained that
°
T-t) ( 1 - C d 2 (x) lIu(t)lIoo ~ u(x, t)
in
n x [to,T)
°
for some constant C > and some to E (0, T), where T is the blow-up time and d(x) = dist(x, on). Therefore,
u(x, t) Ilu(t)lloo
----t
I, as t
----t
T and
{ 0, as t
----t
T
d(x) ;rr;--;
vT-t
and
----t
d(x)
vT-t
----t
00,
° •
In other words, the size of the boundary layer decays like VT - t. Okada and Fukuda59 studied the following scalar equation with both local and localized sources Ut
= D.u + uP(x, t) + uq(x*, t), (x, t)
u(x, t) {
=
0,
u(x,O) = uo(x),
E B x
(0, T),
(x, t) E oB x (0, T), x
E
13,
where B = {x E ]RN : Ixl < I}, x* E B is a fixed point and parameters p, q > 0. The initial data Uo is a C 2 radially symmetric and decreasing positive function. For the special case: x* = 0, under the condition on the initial data uo(x): there is a positive constant I-" such that
D.uo + ub
+ uZ(x*) 2:: 1-",
V x E B,
(23)
they proved that a single point blow-up occurs when p > q + 1 , while total blow-up occurs when p ~ q+ 1. Moreover, the blow-up rates were also obtained. For the general case, Okada and Fukuda59 proved that if q > max{l,p} the total blow-up may occur. If p > q+ 1 and the condition (23) holds, then the point x* is the only blow-up point. Moreover, the blow-up rates were also obtained.
261
Okada and Fukuda also discussed the more general problem:
+ g(t), (x, t)
E
B x (0, T),
u(x, t) = 0,
(x, t)
E
8B x (0, T),
u(x,O) = uo(x),
x
Ut
=
t,.u + f(u)
{
E
B,
for the following cases: (i) f(u) = e QU , g(t) = e i3u (O,t); (ii) f(u) = uP, g(t) = eu(O,t); (iii) f(u) = eU , g(t) = uq(O,t) .
• Systems For the blow-up set of the problem (7), Zheng and Wang 98 obtained the following results: 1° If m < pq + p + 1 n < pq + q + 1, then both u and v are total p+1 ' q+1 blow-up; pq + p + 1 m(p + 1) - p . 2° If p+1 < m ~ q + 1, n ~ m ' then v IS total blow-up, and there exists initial data such that u is single point blow-up; 3° If pq + p + 1 < m < q + 1 m(p + 1) - p < n < pq + q + 1 then v p+1 , m q+1 ' is total blow-up, and there exist initial data such that u is total blow-up as well; pq+q+1 n(q+1)-q. 4° If q+1
262
blow-up while u remains bounded. The simultaneous blow-up and blow-up profiles for the following parabolic systems with localized sources Ut Vt
tn, = ~v + exp{pu(xo, t) + qv(xo, tn, = ~u + exp{mu(xo, t) + nv(xo,
xE n,
t > 0,
x E n,
t > 0,
xE an, t > 0,
u = v = 0, u(x, 0) = uo(x),
v(x, 0) = vo(x),
(24)
xEn
was studied by Li and Wang 52 , where initial data uo(x) and vo(x) are nonnegative C 1 functions vanishing on the boundary an, Xo E n is a fixed point, parameters p, q, m and n are positive constants. Their results are the following: 1° Suppose that the initial datum (uo, vo) satisfies ~uo(x)
+ emuo(xo)+nvo(xo)
~
0,
~vo(x)
+ evuo(xo)+qvo(xo)
~
° in n,(25)
and u and v blow up simultaneously. Then either (a) p ~ m and n ~ q, or (b) p < m and n < q. 2° Assume that (25) holds. If p ~ m and n ~ q, then u and v blow up simultaneously. 3° Assume that (25) holds, u and v blow up simultaneously. Then the following statements are true, and all limits hold uniformly on the compact subset of n. (i) If p > m and n > q then . 11m
u(x, t)
.,...,---:'----'--:-7
Iln(T - t)1 In particular, t->T
lim
t-T
n- q np - mq'
. 11m
t-T
v(x, t) Iln(T - t)1
.,---:'----'-:-:-
p-m np-mq
n-q ) ( u(xo, t) - np-mq lIn (T - t)1
m)
n - q ( In n - q + -n- I n p -- - np - mq np - mq n - q n - q
,
lim ( v(xo, t) - p-m lIn (T - t)1 ) np- mq
t-T
q p-m (In p - m + -p- I n n--) . np - mq np - mq p - m p - m
=.
(ii) If p > m and n = q then . u(x, t) 1 11m t->T In iln(T - t)1 p - m'
1.
1m
t-T
v(x, t)
1
Iln(T - t)1
q
.,...,--:'---'--:-7
263 Particularly, lim {(p - m)u(xo, t) -In Iln(T - t)l} = In p - m, q
t--+T
=m
(iii) If p
and n > q then
, u(x, t) I1m .,----'--'-'--
lim ~_v(c..,.x..:....'..!..t)___ t--+T In Iln(T - t)1
1
Iln(T - t)1
t--+T
m'
1
n-q
Particularly,
= In n - q,
lim {(n - q)v(xo, t) -In Iln(T - t)l}
m
t--+T
(iv) If p
= m and n = q then
, I1m t--+T
(v) If p
u(x, t) Iln(T - t)1
1
-m n'
+
v(x, t) , I1m -:---:'---'---'-,...,. Iln(T - t)1
p+q
v(x, t) , I1m ---'---'----'-Iln(T - t) I
m-p mq-np
1
t--+T
< m and n < q then
, u(x, t) I1m ---'--''-:-Iln(T - t)1
t--+T
q-n mq-np'
t--+T
In particular, lim
t--+T
(
u(xo, t) -
q-n Iln(T - t)1 ) mq - np
= q-n (Inq - n mq - np
mq - np
q-n) + -n- I n -q- n m- p ,
lim (V(x o, t) -
t--+T
m - p Iln(T - t)l) mq - np p p m= m-p (Inm + -p- I n --) , mq - np mq - np m - p q- n
Li and Wang 53 also considered the following semilinear parabolic system with localized reaction terms Ut
Vt
+ u<>(xo, t)vP(xo, t), = ~v + uq(xo, t)vi3(xo, t),
x E
t
> 0,
x
n, E n,
t
> 0,
x
E
an,
= uo(x), v(x,O) = vo(x), x
E
n,
= ~u
u = v = 0, u(x,O)
t
> 0,
(26)
where a, /3, p, q are nonnegative constants and satisfy a + p > 0, /3 + q > 0, The initial data uo(x), vo(x) E Co(n) and are non-negative nontrivial, Xo E n is a fixed point,
264
For the simultaneous blow-up they obtained the following results: 1° Assume that (u, v) is the classical solution to (26) in n x (0, T), u and v blow up simultaneously in the finite time T. Then the parameters a, (3, p and q must satisfy: (a) p 2: (3 -1 and q 2: a -1, or (b) p < (3-1 and q < a-1. 2° Assume that (u, v) is the classical solution to (26) in n x (0, T), which blows up in the finite time T. If p 2: (3 - 1 > and q 2: a-I> 0, then u and v must blow up simultaneously. For the uniform blow-up profiles in the interior, they obtained 3° Assume that (u, v) is the classical solution to (26) in n x (0, T), u and v blow up simultaneously in the finite time T. Then the following statements hold uniformly on any compact subset of n. (i) If (a) p> (3-1, q > a-I and pq > (a-1)«(3-1), or (b) p < (3-1 and q < a-I, then
°
lim u(x, t)(T - t)e = lim lIu(t)II=(T - t)e = {-Ol (o/a)p/(p+l-f3)}-e , t->T
t->T
{ lim vex, t)(T - tt = lim IIv(t)11 t->T
t->T
=
I (T - t)<7 = { -(a jO)q/(q+l-a) a
}-<7
where
0=
q+1-a p+1-(3 a=-------;-------:--:-::----:pq - (a - 1) «(3 - 1) , pq - (a - 1) «(3 - 1) .
(ii) If p > (3 - 1 and q = a-I> 0, then lim ua-1(x, t)[ln u(x, t)]p/(P+l-{3)(T _ t)
=
t->T
vp +1 - f3 (x, t) t->T Iln(T - t)1 .
lIm
(iii) If p = (3 - 1 >
p
a-I
lim { lim t->T
°
and q
q + 1 - (3 (3 - 1 '
= -=-----'-
lim v f3 - 1(x, t)[lnv(x, t)]q/(q+l-a)(T _ t) =
t->T
(3
and q > a-I, then
. uq+l-a(x, t) hm Iln(T - t)1
(iv) If p = (3 - 1 >
+1-
= -'-----
°
t->T
t->T
1 (a - 1)(p + 1 - (3)p/(p+l-f3) '
1 . «(3 - 1)(q + 1 - a)q/(q+l-a)
= a-I > 0, then we have
u(x, t) = lim Ilu(t)ll= = _1_ Iln(T - t)1 t->T Iln(T - t)1 p + q' vex, t) = lim Ilv(t)II=1 = _1_ Iln(T - t)1 t-+T In(T - t)1 p + q'
265
For the boundary layer size, they obtained the following results: 4° Under the condition (i) of 3° then, for all K > 0, there exist some constants C 2 ~ C 1 > and some to E (0, T), such that
°
d(x) d(x) C 1 ~llu(t)lloo:::; u(x,t):::; C 2 ~lIu(t)lloo, yT-t yT-t d(x) d(x) { C 1 jT=t 11 v(t)lI oo :::; v(x,t) :::; C 2 jT=t 11 v(t)lI oo T-t T-t for all (x, t) E n x [to, T) such that d(x) :::; K vT - t, where d(x)=dist(x, an). This result shows that near the blow-up time in the boundary layer, u and v behave like lIu(t)lIood(x)/vT - t and IIv(t)lIood(x)/vT - t respectively. Furthermore, one can deduce that the size of the boundary layer is at least of order vT - t near the blow-up time, in the sense that u(x, t) = o(llu(t)lIoo) and vex, t) = o(lIv(t)lloo) as t - t T and d(x)/vT - t - t 0. 5° Under the condition (i) of 3°, if it is further assumed that e > 1 and a > 1. Then there exist constants C > and to E (0, T), such that, in n x [to, T),
°
u(x,t)
~ (1-C~2(xD Ilu(t)lIoo,
v(x,t)
~ (1-C~2(xD IIv(t)lloo.
It follows from 4° and 5° that
u(x, t) lIu(t)lIoo
-t
{ u(x, t) lIu(t)lIoo
-t
1,
° ,
vex, t) Ilv(t)lloo
-t
vex, t) IIv(t)lloo
-t
1
as t
° as
t
-t
d(x) T and jT=t T-t
- t 00,
-t
T and v d(x) T _t
-t
° .
This shows that the size of the boundary layer is exactly of the order vT - t. In other words, the size of the boundary layer decays like vT - t. A special form of the system with k components and localized terms: Uit =
Ui {
i
6.ui
+ Uf+l (xo, t),
= 0,
= 1,· ..
,k,
Uk+l
x E
n, t > 0,
x E
an,
t > 0,
= Ul
was discussed by Pedersen and Lin 63 . Some results on the blow-up rate estimates and blow-up profiles were obtained.
266 Recently, Li and Wang 54 considered the following semi-linear parabolic system with nonlinear localized sources
+ um(x, t)vn(xo, t), flv + uP(xo, t)vq(x, t),
Ut = flu
xED, t
Vt =
xED, t > 0,
u
= v = 0,
x
> 0, (27)
aD, t > 0,
E
u(x, 0) = uo(x), v(x, 0) = VO(x), xED, where parameters m, n, p, q are nonnegative constants and satisfy m+n > 0, p+q > 0. Initial data uo(x), vo(x) E Co(D) and are non-negative. Xo ED is a fixed point. The following results were obtained. 1° Assume that (u, v) is the classical solution of (27), which blows up in finite time T. Let m ::; 1, q ::; 1, then the following statements hold uniformly on any compact subset of D. (i) If m < 1, q < 1 and np - (1 - m)(1 - q) > 0, then
1~Ilj. u(x, t)(T - t)0 = (;8 (~)
np
(1 ';,,)(1
q) ,
p
(1 m)(l
( ) ) np
= aU -
lim v(x, t)(T - t)U
t-->T
q)
a
(
,
where
()_ l+n-q - np- (l-m)(l-q)'
l+p-m
a
= -np---:-(l-:-_--=---m--:-)--:-c(l-----:-q)'
(ii) If m = 1 and q < 1, then In u(x, t) 1+n - q . I1m ---t-->T 11n(T - t)1 np ,
lim In v(x, t) =~. Iln(T - t) I n
t-->T
(iii) If m = q = 1, then In u(x, t) . 11m --'---'---'--,. Iln(T - t)1
t-->T
(iv) If m < 1 and q
= 1,
,
In v(x, t) . 11m -:-:--=':,--'----'7-: Iln(T - t) I
t-->T
1 n
then
lim lnu(x,t) =~, I1n(T - t)1 p
t-->T
1 p
lim lnv(x,t) jln(T - t)1
t-->T
l+p-m np
For the special case: m 2: 1, q 2: 1, and D = B(O; R), Xo = 0. Initial data Uo, vo B(O; R) --+ Rl are nonnegative and radially symmetric nonincreasing continuous functions and varnish on aB(O; R), and satisfy fluo(x) + uQ1(x)v~(O) 2: and flvo(x) + ug(O)vZ(x) 2: in B(O; R),
°
°
267 Li and Wang obtained the following results: 2° Assume that (u, v) is the classical solution to (27) in B(O; R) x (0, T), which blows up in finite time T. If p ~ m - 1 > 0 and n ~ q - 1 > 0, then u and v must blow up simultaneously; 3° Assume that (u, v) is the classical solution to (27) in B(O; R) x (0, T), u and v blow up simultaneously in finite time T. Then the parameters m, n, p and q must satisfy: (a) p ~ m - 1 and n ~ q - 1, or (b) p < m - 1 and n < q -1; 4° Assume that (u, v) is the classical solution to (27) in B(O; R) x (0, T), u and v blow up simultaneously in finite time T. Then there exist constants 0< c ~ C such that the following statements hold for all 0 ~ t < T. (i) If (a) p > m - 1 and n > q - 1, or (b) p < m - 1 and n < q - 1, then
c(T - t)-e ~ u(O, t) ~ C(T - t)-e, c(T - t)-a ~ v(O, t) ~ C(T - t)-a, where the constants () and (j were given in l°(i). (ii) If p > m - 1 and n = q - 1 > 0, then c < uHp-m(O, t) < C c < vn(O t){lnv(O t)}p/(HP-m)(T - t) < C. - Iln(T - t)1 - , -" -
(iii) If p = m - 1 > 0 and n > q - 1, then
vHn-q(O t) c ~ uP(O, tHIn u(O, t)}n/(Hn-q)(T - t) ~ C, c ~ Iln(T _ ~)I ~ C. (iv) If p = m - 1 > 0 and n
=q-
< Inu(O,t) < c - Iln(T _ t) I -
c,
1 > 0, then In v(O, t)
c
C
~ Iln(T - t)1 ~ .
Later, Deng,24 Du,25 Wang and Wei 88 generalized the problem (27) to the quasilinear case:
En x (0, T), (x, t) E n x (0, T), (x, t) E an x (0, T),
Ut
=
uO:(.6.u + uP(x, t)vq(xo, t)), (x, t)
Vt
=
vi3 (.6.v
+ vm(x, t)un(xo, t)),
u(x, t) = v(x, t) = 0,
(28)
u(x,O) = uo(x), v(x,O) = vo(x), x E Q, where parameters q,n > 0, p,m ~ 0, a,(3 E (0,1). Under the suitable conditions on the initial data uo(x) and vo(x), the similar results to that of54 for the blow-up rates and uniform blow-up profiles were obtained. When the domain n is a ball and Xo = 0, it was proved in the paper88 that
268 x = and
°is the only blow-up point. Moreover, if ° 1 such that
p, m> 1 and there exist c >
°
~ ~ ~
Then
u(r, t)
~
Cr-'Y1, vCr, t)
hold for some constant C >
~
Cr-'Y2,
(r, t) E (0, R] x [0, T)
°
and any,l > 2/(p - 1),
,2 > 2/(m - 1).
6. Degenerate problems not in divergence form The following degenerate problem not in divergence form Ut
+ au), x E n, t > 0, uo(x), x E n,
= uP(~u
u(x,O) = {
u(x, t) = 0,
(29)
x E 8n, t > 0,
has been discussed by many authors, see Chen9 , Friedman and McLeod 33 , Gage 35 , Passo and Luckhaus62 , Wang et a192 , Wiegner95 and Zimmer 101 . Let ),1 be the first eigenvalue of -~ in n with homogeneous Dirichlet boundary condition. When p = 2, Friedman and McLeod 33 proved that if ),1 > a then the solution u of (29) exists globally, while if ),1 < a then the solution u of (29) blows up in finite time. Gage 35 considered the blowup case again for p = 2, and gave an estimate for the blow-up time and made a more detailed study of the blow-up set. For the case ),1 > a and p > 2, Zimmer in 101 studied the global existence of solution to (29). For the case 1 < p < 2, Chen9 proved that the solution of (29) blows up in finite time for the large domain n (meaning ),1 < a) and large initial data. Wiegner 95 considered the problem (29) for the case p > 1 and proved that the solution of (29) blows up in finite time if ),1 < a. Moreover, he obtained the uniqueness of solution when p 2 2. Wang et. a1 92 discussed (29) for the case p > 1. At first, they obtained the uniqueness result, and then proved that all positive solutions of (29) exist globally if and only if ),1 2 a. Passo and Luckhaus62 studied the properties of solutions to (29). Later, some authors generalized the problem (29) to systems. Wang and
269 Xie89 studied the following problem
Ut = vP(b.u + au),
n, t > 0, Vt = uq(b.v + bv), x E n, t > 0, u(x,O) = uo(x), v(x,O) = vo(x), x E n, u(x, t) = v(x, t) = 0,
x
E
x EOn, t > 0,
where p, q, a and b are positive constants, and p, q uo(x) and vo(x) satisfy 1-
UO, Vo E C (n), Uo > 0, Vo > 0, {
Uo
auo
(30)
°
= Vo = 0, aTJ < 0, avo aTJ < ,
~
1. The initial data
x E
n,
xE
U~~.
!:In
They obtained the following results: 1° When min{a, b} ~ AI, then the problem (30) has at least one global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (to our knowledge, up to now the uniqueness is an open problem); 2° When min{a, b} > AI, then the problem (30) has no global positive classical solution. If in addition the initial data (uo, vo) satisfies b.uo+auo ~ 0, b.vo + bvo ~ in n, then the positive classical solution is unique and blows up in finite time. Wang 81 studied an another form of the problem (30):
°
Ut = uP(b.u + av),
x E n, t > 0,
Vt = vq(b.v + bu),
x E n, t > 0,
u(x,O) = uo(x), v(x,O) = vo(x), xE n, u(x, t) = v(x, t) = 0,
xE
an,
(31)
t > 0.
It was proved that all positive classical solutions exist globally if and only ifab~Ai·
Later, Li and Xie 56 generalized the above problem to the systems with several components. Recently, Chen and Wang 12 generalized the problem
270 (31) to the following problem with nonlocal sources:
l ~v + l
Ut
= f (u) ( ~ U + a
Vt
= g(v) (
b
XED, t > 0,
VdX) ,
XED, t > 0,
UdX) ,
u(x, t)
=
x
E
aD, t > 0,
U(X,O)
= uo(x), v(x,O) = vo(x), x
E
D.
vex, t)
=
0,
Assume that the following conditions hold: (HI) uo(x), vo(x) E C 2 +<>(D) n C(n) for some uo(x), vo(x) > in 0.;
°
(H2)
uo(x)
= vo(x) = 0,
~~o
°
Either liminf fees)) > s ..... oo g s
~v;
< 0,
outward normal vector on aD; (H3) f,g E C[O, 00) nC 1 (0, 00), f(O) in (0,00); (H4)
(32)
<
°<
< 1,
a
°
on aD, where
Tj
= g(O) = 0, and f, g > 0,
is the
/ ,g' ~
°or liminf fg((s)) > °holds. s ..... oo
S
They obtained the following results: 10 The solution (u, v) of (32) exists globally if and only if one of the following holds:
(i)
ab::;~, J.L
ds ... ) 100 sg(s) ds (.. ) 100 sf(s) = 00, ( 111
11
= 00.
20 Assume that the solution (u, v) of (32) blows up in finite time T and
00
1
00
ds
f(s)
=
00,
1
ds
g(s) = 00.
Then (u, v) blows up globally, where J.L
=
l
and
= 1, xED;
x E aD.
We mention that Chen and Goo l l studied the single equation case of
271
(32) with weighted function:
Ut
f(u) ( Llu + a(x)
=
l
UdX) , XED, t > 0,
1
aD, t > 0,
u(x,t) = 0,
x
u(x,O) = uo(x),
xED,
E
where the weighted function a(x) is continuous, bounded and positive.
7. Non-local problems
Many physical phenomena were formulated into non-local mathematical models and studied by many authors, see for example, Bebernes and Bressanl, Bebernes and Eberly 3, Chadam et al 8 , D ay 15,16, Deng et al 19 ,22, Friedman 31 , Guo and Su 40 , Pao60 , Souplet68 ,69, and Wang and Wang87 . Bebernes and Bressan 1 studied an ignition model for a compressible reactive gas which is a non-local reaction-diffusion equation. Beberbes et al 2 considered a more general model
Ut - Llu {
f(u)
=
+ g(t),
xED, t > 0,
u(x,O) = uo(x),
xED,
u(x, t) = 0,
x
where uo(x) :::: 0, g(t)
>
0 or g(t) =
E
(33)
aD, t > 0,
1~ll Ut(x, t)dx
with k
>
O.
Chadam et al 8 studied another form of (33) with f(u) = 0 and g(t) = 'l/J(u(x, t))dx, and proved that the blow-up set is the whole region (including the homogeneous Neumann boundary conditions). Souplet 68 ,69 considered (33) with the general function g(t). Pao60 discussed a non-local reaction-diffusion equation arising from the combustion theory. Wang et a1 87 studied the following problems
In
Ut - dLlu =
l
uq(x, t)dx - ku P , xED, t > 0,
u(x,O) = uo(x), u
= 0,
or
au
a", = 0,
xED,
x E
aD, t > 0,
(34)
272
and Ut -
= uo(x),
u(x,O)
1 ~~
x
dtlu = 0,
=
x
fan uq(x, t)dS -
ku P , x
n, E n, E
Eon,
t
> 0,
t
> 0.
Conditions on the global existence and finite time blow-up of solution were obtained. Moreover, it was proved that the blow-up set for the problem (34) is the whole domain The following three maximum principles are useful in the study of nonlocal problems. Maximum Principle87 Let d be a positive constant, and a, (3 be continuous and nonnegative functions and satisfy a+(3 > 0, Ci be continuous and bounded functions and C2 ~ 0. Ifu E Cl,O(QT)nC 2,1(QT) and satisfies
n.
Ut -
dtlu
a(x, t)u
~ Cl(X, t)u +
au
+ (3(x, t) ary
k
C2(Y, t)u(y, t)dy, (x, t) E QT,
~ 0, (x,
u(x,O) ~ 0,
t) EST,
x E
n.
°
Then u ~ 0 on QT. Moreover, u > in QT provided that u(x, 0) =1= 0. Here QT = x (O,T] and ST = x (O,T]. Maximum Principle87 Let u E C(QT)nC 2,1(QT)nC 1,O(n x (0, T]) and satisfy
n
an
Ut -
dtlu ~ 0,
(x, t) E QT,
~~ ~ Cl (x, t)u + fa_n C2(y, t)u(y, t)dSy,
1
u(x,O) ~ 0,
x E
(x, t) EST,
n.
°
°
If ci are continuous and bounded, i = 1, 2, and C2 ~ on ST. Then u ~ on QT. Moreover, u> in QT provided that u(x, 0) =1= O. Maximum Principle84 Let functions ai, b be continuous on QT, the function C be nonnegative and bounded in QT, functions f and 9 be nonnegative and bounded in QT and an x QT, respectively. If u E
°
273
C (QT ) n C 2,1 (QT) and satisfies
Ut - 6u U
~
l
~ aiDiu + bu + f(x, t)
l
c(y, t)udy, (x, t) E QT,
g(x, y, t)u(y, t)dy,
(x, t)
u(x,O) = uo(x) ~ 0, then we have u ~
x E
°
on QT.
For the special case: ai = b = f
°
==
and 9 satisfies
l
EST,
n, g(x, y, t)dy < 1
for all (x, t) E 80, x [0, T], the last maximum principle was given by Deng 18 firstly. The ignition period of a thermal event can be described by the following integra-parabolic problem (Bebernes et al 2 , Benernes and Eberly 3, Bricher4 ,5 )
Ut = 6u + eau +
1'1~ll
l
ei3u dx, x E 0"
u(x,t) = 0,
1
u(x,O)
=
t > 0, (35)
x E 80" t > 0, ~
uo(x)
x
0,
E 0"
where a and f3 are positive constants, u is the temperature perturbation of the gas, and l' > 1 is the gas parameter. In case that a = f3 = 1, Bricher 5 proved that the solution blows up everywhere if N ::; 2, and blows up at a single point if N ~ 3 and l' - 1 > is sufficiently small. Recently, Liu and Wang 58 studied the uniform blow-up profiles of blow-up solutions to the problem (35) when 0, = {x E]RN : Ixl < R}. Assume that the initial data Uo satisfies
°
(AI) Uo E C 2 (0.), Uo = uo(r) with r = lxi, uo(r) R, uo(R) = and u~(O) < 0.
°
(A2) 6uo
+ eauo + 1'1~ll
l
ei3uo dx
~ 0,
<
° °< for
r ::;
x En.
They obtained the following results: 1° Let
u(x, t) Iln(T* - t)1
=0,
v x E n \ {O};
(ii) if N 2:: 3, then the solution u blows up only at the single point x and satisfies u(O, t) 1 . 11m t--->T* Iln(T* - t)1 a
=0
3° Assume that the assumptions (AI) and (A2) hold. If a > 13, then the solution u blows up only at the single point x = 0 and satisfies
. 11m t--->T*
u(O, t)
1
Iln(T* - t)1
a
45
Recently, Kong and Wang considered the following semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries
in + in
Ut
= flu +
um(x, t)vn(x, t)dx, x E
n,
Vt
= flv
uP(x, t)vq(x, t)dx,
x E
n, t > 0,
x E
an,
t > 0,
7jJ(x, y)v(y, t)dy,
xE
an,
t > 0,
u= v=
in in
u(x,O) = uo(x), v(x,O) = vo(x), x E
t > 0,
(36)
n,
where m, n, p and q are nonnegative constants and satisfy m+n > 0, p+q > O. Functions
an
m, q:S 1 and np:S (1 - m)(1 - q).
275
Then the solution exists globally. 2° Assume that one of the following conditions holds:
(a) m> 1; (b) q> 1; (c) np> (1- m)(1 - q). Then the solution blows up in finite time for the sufficiently large initial data. 3° Assume that
In
In
(a) m> 1; (b) q> 1; (c) np> (1 - m)(1 - q). Then the solution blows up in finite time for any positive initial data. 4° Assume that
10
E
an.
If the solution blows up in finite time, then the blow-up set is the whole domain
n.
5° Assume that
10
E
an,
and (u, v) is the classical solution and u and v blow up simultaneously in finite time T. Then the parameters m, n, p and q must satisfy: (a) p ~ m-1 and n ~ q - 1, or (b) p < m - 1 and n < q - 1. 6° Assume that
10
and (u, v) is the classical solution and blows up in finite time T. If p ~ m - 1 > 0 and n ~ q - 1 > 0, then u and v must blow up simultaneously. 7° Under the conditions of 5°, the following limits hold uniformly on any compact subset of n: (i) if (a) n > q - 1, P > m - 1 or (b) n < q - 1, P < m - 1, then lim u(x, t)(T - t)6
t-+T
= lim Ilu(t)lloo(T - t)6 t-+T
C~I (~r/(n+l-q») -6,
=
lim v(x, t)(T - t)lT
t-+T
= lim Ilv(t)lloo(T - tt t-+T
=
C~I (~r/(p+l-m») -a,
where
e-
n+1-q - np - (m - 1)(q - 1)'
p+1-m • - np - (m - l)(q - 1)'
(J-
(37)
276
(ii) if p
=m-
1>
°and n >
> 0, then
q- 1
E.Ilj.uP(x,t)[lnu(x,t)]n/(n+l-q)(T-t) = lim v n+1-q(x, t)lln(T _ t)I-1 t--+T
(iii) if p> m - 1 >
(n + 1
q)-n/(n+l- q)
- plnl
'
= n +1- q;
°and n = q - 1 > 0, then
P
lim u P+1- m (x, t)lln(T _ t)1- 1 = P + 1 - m, n + (p 1 m)-p/(p+l-m) }~Ilj.vn(x,t)[lnv(x,t)]P/(P+1-m)(T-t) = - nlnl ; t--+T
(iv) if p = m - 1 >
°and n =
lim Inu(x,t)1 In(T - t)1
t--+T
{ lim In v(x, t) t--+T Iln(T - t)1 Especially, if
q- 1
> 0, then
= lim Illnu(t)lloo _ _1_ t--+T
Iln(T - t)1 - n
= lim IlIn v(t)lloo = t--+T
IIn(T - t)1
+ p'
(38)
_1_
n
+ p'
10 cp(x, y)dy = 1 and 10 '¢(x, y)dy = 1 for all x
E an, then
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282
GLOBAL SOLUTIONS FOR THE SHIGESADA-KAWASAKI-TERAMOTO MODEL WITH CROSS-DIFFUSION Yoshio Yamada* Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
1. Problems
In this article we will discuss the existence of global solutions to the following system appearing in population biology: Ut
Vt
+ av + /u)u] + au(l - u = d2~[(1 + (3u + <5v)v] + bv(l - du -
= dl~[(l
Bu
=
Bv
=
°
u( . ,0) = Uo 2:: 0, v(·, 0) = Vo 2::
°
cv)
in
v)
in
nx nx
on
an x (0, (0),
in
n,
(0,00), (0,00),
(1.1)
where n is a bounded domain in RN (N 2:: 1) with smooth boundary Bu = u (Dirichlet boundary condition) or Bu = au/av with outward normal v on an (Neumann boundary condition); a, (3, /, <5 are nonnegative constants and d 1 , d 2 , a, b, c, d are also positive constants. The system (1.1) is known as the Lotka-Volterra competition system with nonlinear diffusion effects. In (1.1), u and v, respectively, represent the population densities of two competing species which are interacting and migrating in the same habitat n. Such a population model was first proposed by Shigesada, Kawasaki and Teramoto [20] in 1979 to study pattern formation phenomena between two competing populations. According to Okubo and Levin [16], the formulation of diffusion effect is based on the assumption that individual species move under the
an;
*Part of this work has been supported by Grant-in-Aid for Scientific Research (C), No. 18540223, Japan Society for the Promotion of Science.
283 influence of the following forces; (i) a dispersive force associated with the random movements of individuals; (ii) an attractive force, which induces directed movement of individual species toward favorable environments; and (iii) population pressure due to interferences between individual species. In (1.1), d i (i = 1,2) represent random dispersive force of movement of an individual and a, (J, " 0 describe mutual interferences between individuals due to the population pressure. Especially, a and (J are usually referred as cross-diffusion coefficients, while , and 0 are referred as self-diffusion coefficients. For detailed biological background of the nonlinear diffusion terms, see the monograph of Okubo and Levin [16]. Note that, in case Bu = u for (1.1), the boundary condition means that the habitat 0 is surrounded by a hostile environment. In case Bu = aujav, the boundary condition aujav = 0 is usually called no-flux condition, which implies that there is no migration across the boundary. Many numerical simulations for (1.1) exhibit interesting pattern formation. So the analysis of this system and related ones is very important from the mathematical view-point as well as the ecological view-point. When local solvability is concerned, Amann [1] proved that (1.1) has a unique local solution for any uo, Vo E (0) by using the theory of quasilinear evolution equations. We can also apply the local existence results in the monograph of Ladyzenskaja, Solonnikov and Ural'ceva [8] if classical solutions of (1.1) are concerned. In case N = 1, Kim [7] proved the existence of a unique global solution of (1.1) for any nonnegative initial data. (See also the work of Shim [21], where the uniform boundedness of solutions for (1.1) is shown in case N = 1 by putting some restrictions on nonlinear diffusion coefficients.) After that, a lot of mathematicians have attempted to show the global solvability of (1.1). For any initial data and space dimension, Deuring [5] proved that, if both a and (J are small depending on the amplitude of initial data and, = 0 = 0, then (1.1) has a unique global solution. However, the analysis is very hard in a higher dimensional case when we put no restrictions on cross-diffusion coefficients a, (J. In case N = 2, Yagi [25-27] established the existence of global solutions of (1.1) by assuming one of the following conditions:
W;
(A.l) 8dn > d 2 (J > 0 and 8d 2 o > dla > 0, or (A.2)
,> 0 and
(J
= 0 = o.
His global result reads as follows; if nonnegative functions uo, Vo satisfy H1+€(O) with some € > 0, then (1.1) admits a unique nonnegative
uo, Vo E
284
global solution u, v satisfying
u, v E C( (0,00); H2(n»
n C 1((0,00); L2(n)).
The above assumptions (A.l), (A.2) imply that self-diffusion is dominant over cross-diffusion in a sense. Here it should be noted that (A.l) can be weakened in the following form (A.l), 64,0' > 0'.(3 or 64,5
= 0'.(3 >
°
to assure the global existence (see the work ofIchikawa and the author [6]). In these works, positivity of self-diffusion coefficients has played a central role in getting suitable a priori estimates of solutions of (1.1). Lou, Ni and Wu [14] have studied the case where self-diffusion coefficients in (1.1) may be absent. Using Amann's local existence result for N = 2, they have shown that, if (3 = and p > 2, then (1.1) has a unique global solution u, v E C( (0,00); W~ (n» for any nonnegative initial functions uo, Vo E W~ (n). The purpose of the present article is to discuss the problem on the existence of global solutions of (1.1) in case N ~ 3 without any restrictions on the amplitude of initial functions Uo, Vo. However, we have very few results to this problem for the fully nonlinear diffusion system (1.1). In what follows, we will focus on the following system
°
(P)
+ O'.v + ,u)u] + au(1 - u d2~[(1 + 5v)v] + bv(1 - du - v)
Ut
= d1~[(1
Vt
=
au=av=o av av u( . ,0) = Uo ~ 0, v(·, 0) = Vo ~
°
cv)
in III
nx nx
(0,00), (0,00),
on
an x (0,00),
in
n,
which is slightly simpler that (1.1) because diffusion in the second equation of (P) does not contain a cross-diffusion term. In the first equation we assume 0'.>0
and
(1.2)
Global existence results for (P) are stated in a different style according as 0' = or <5 > 0. Roughly speaking, if <5 = 0, then (P) possesses a unique global solution for any initial functions uo, Vo and any space dimension N ~ 3; while, if 5 > 0, we have to put some restrictions on the space dimension or nonlinear diffusion coefficients to assure the existence of a global solution of (P) for any initial functions Uo, Vo. We will give global existence results and explain some ideas and methods to show how these results can be proved.
°
285
The contents of the present paper are as follows. In §2 we will state local and global existence results of solutions of (P). In §3, we will give a priori estimates of solutions of (P), which are helpful to show that local solutions can be extended for all t E [0,00). The ideas and methods of analysis to prove the global existence results will be briefly stated in §4. Finally, some concluding remarks are given in §5. We should note that most of the results in this paper are valid with some modification if zero Neumann boundary conditions in (P) are replaced by zero Dirichlet boundary conditions.
2. Global existence results We begin with local existence results for (1.1), which are valid for any space dimension. The following theorem has been established by Amann [1].
Theorem 2.1. Assume that nonnegative initial data (uo, va) satisfies Uo, Vo E Wt(D) with p > N. Then (1.1) possesses a unique solution (u, v) satisfying u, v E C([O, T); W~(D)) n C((O, T); W;(D)) n C 1 ((0, T); Lp(D)), where T > 0 is a maximal existence time. Moreover, if (u, v) satisfies sup lIu(t)lIwl(fl) < 00
O~t
and
p
sup IIv(t)lIwl(fl) < 00,
09
p
then T = 00. Theorem 2.1 is useful if we work in the framework of LP(D) spaces. If classical solutions of (1.1) are concerned, we have to use the following result (see [8]):
Theorem 2.2. Assume that nonnegative initial data (uo, va) satisfies Uo, Vo E c2+>'(11) with A > 0 and ou%v = ov%v = 0 on aD. Then (1.1) possesses a unique solution (u, v) E C 2+>',(2+>')/2(11 x [0, T)), where T > 0 is a maximal existence time. By virtue of Theorems 2.1 and 2.2, it is sufficient to show some suitable a priori estimates of (u, v) in order to establish the global existence to (1.1). In what follows, we will mainly study (P). For the sake of simplicity, we assume that initial data (uo, vo) satisfies (A)
Uo ~ 0, Vo ~ 0 in D, with
Uo, Vo E c2+>'(11)
_ avo >. > 0 and ouo ov - avon
~"
ua G.
286 The first global result is concerned with the case where the diffusion term in the second equation of (P) is linear; 0 = O. We have the following theorem due to Choi, Lui and Yamada [3, 4].
Theorem 2.3. For 0 = 0, assume that (uo, va) satisfies (A). Then (P) possesses a unique solution (u, v) such that u,v E C2+>.,(2+>.)/2([2 x [0,00)). Theorem 2.3 implies that, if a self-diffusion coefficient is positive in the first equation and diffusion is linear in the second equation in (P), then the global existence result holds for any space dimension. The same result is proved by Wang [24] for (P) with reaction terms replaced by slightly more general ones. The second result is concerned with case 0 > 0 in (P). One of the global existence results is given by the following theorem proved in [4].
Theorem 2.4. Let 0 > 0 and N :::; 5. If (uo, va) satisfies (A), then (P) possesses a unique solution (u, v) satisfying u, v E C2+>.,(2+>')/2 ([2 x [0,(0)). Remark 2.1. In [11], the same restriction N :::; 5 is also imposed to derive the global existence result. Recently, the restriction on the space dimension in Theorem 2.4 has been weakened by Th9c [23, Theorem 1.1.2] as follows.
Theorem 2.5. Let 0 > 0 and N :::; 9. Then the same conclusion as Theorem 2.4 holds true. Both Theorems 2.4 and 2.5 have restrictions on the space dimension. Another global existence result is also shown by Th9c [22].
Theorem 2.6. Let 0 > 0 and N::::: 1 and let (uo,vo) satisfy (A). Assume (i) d1Ct < 2d 2o or (ii) d1Ct = 2d2o and d 1 :::; d 2 . Then (P) admits a unique solution (u,v) satisfying u,v E C2+>.,(2+>')/2([2 x [0,(0)). Theorem 2.6 is a generalization of the work of Le and Nguyen [12], where some additional conditions are imposed on nonlinear diffusion coefficients. Although there is no restriction on the space dimension in Theorem 2.6, a self-diffusion coefficient in the second equation must be sufficiently large compared to a cross-diffusion coefficient in the first equation to assure the global existence.
287 Finally we should state a global existence result for (1.1) obtained by
Le [10]. Theorem 2.7. Let 8 > 0" > 0 and N ;::: 1 and let (uo, vo) satisfy (A). Assume dia < d 28 and d 2(3 < dn. Then (P) admits a unique solution (u,v) satisfying u,v E C2+>.,(2+>')/2(n x [0,00». 3. A priori estimate In this section we will give some a priori estimates for solutions of (P). We begin with the following lemma.
Lemma 3.1. Let (u, v) be a solution of (P) in [0, T]. Then u ;::: 0
nx
where QT = Iw(x)loo.
and
m;::: v
[0, T] and m =
;::: 0
in QT,
max{l,
Ilvolloo}
with
Ilwlloo .-
SUPxEO
Proof. The first equation in (P) is expressed as Ut
= d l (l + av + 2,u)~u + 2d I (a\7v + ,\7u) . \7u +
{adl~v
+ a(l- u -
(3.1)
cv)}u
and the second one is written as Vt
= d2 (1 + 28v)~v + 28d2 \7v· \7v + b(l- du - v)v.
(3.2)
Then application of the maximum principle to (3.1) and (3.2) yields the nonnegativity of u and v (see, e.g., [18]). Applying the maximum principle to (3.2) again one can also show the boundedness of v. 0
Lemma 3.2. Let (u, v) be a solution of (P) in [0, T]. Then and allull£2(Qr):::; Il u oIlLl(o)e
sup Ilu(t)llv(o):::; Iluollv(o)eaT
aT
09~T
Proof. Integration of the first equation in (P) with respect to x gives
! In
u dx
= dl =dl :::; a
In ~[(1 + {
av
+ ,u)u]dx + a
~[(l+av+,u)u]dO'+a
Jao 8v
In
In
u dx - a
In
2
u dx.
(1 - u - cv)u dx {(1-u-cv)udx
Jo
.
288 Hence Gronwall's inequality yields
for
0::; t ::; T;
o
whence the conclusion easily follows at once.
We are now ready to derive the following fundamental estimates for u.
Proposition 3.1. Let T > 0, q > 1 and let (u, v) be a solution of (P) in
[O,T]. If either (i)
<5
=
0 and q
> 1 or (ii)
<5
> 0 and
2~ ~21) > q > 1,
then there exists a positive constant CT satisfying
Proof. We will give the proof in case <5 = O. See [4] for the proof in case <5 > O. We multiply the first equation in (P) by u q - 1 to get:
~i q dt
r u q dx= inr uq-1Ut dx
in
=d 1
In In
q u - 1V[(1+av+21'u)Vu] dx+d1a
+a
In
uq-1V[uVv] dx
q u (1-u-cv) dx
=-(q-1)d 1 +a
In
In
(3.3)
(1+av+21'u)uq-2IvuI2dx-(q-1)dla
In
uq-1vu· Vv dx
q u (1-u-cv) dx.
Set
It
=
In
(1
+ av + 21'u)u q- 2IVuI 2 dx,
q 13= InU (1-U-Cv)dX.
12 =
-In
uq-1Vu· Vv dx,
289 Since u and v are nonnegative by Lemma 3.1, it is easy to see
h ~ 2, ku ~
13
q
-
1V'u I2 dx =
1
k
u Q (l-u)
dx~
(q
!'1)2 kIV' (u(Q+1)/2) 12 dx,
Inl,
where Inl denotes the volume of n. We also note
Therefore, one can deduce the following inequality after integration of (3.3) with respect to t:
for 0
~
t
Ilu(t)lllq(o)
+ collV'(u(Q+1)/2)lli2(Q,)
~
+ aqlnl + (q -
~
Iluolllq(o)
1)d1a
r u Q6.v dxdt JQT
(3.4)
T with a positive constant Co. By Holder's inequality
We now invoke the maximal regularity theory for parabolic equations (see, e.g., Amann [2]). It follows from the second equation in (P) that II V tIILq+l(QT)
~
+ II6. v IlLQ+l(QT)
M (1Ivollw';+l(O)
~ C1 (1
+ Ilbv(l- du -
v)IILQ+l(QT))
(3.5)
+ lIuIlLQ+l(QT))
with some positive numbers M and C1 . Here we have used (A) and Lemma 3.1. Hence we see from these inequalities (3.6)
with a positive constant C 2 > O. Substitution of (3.6) into (3.4) enables us to derive
290
with C 3 > O. We introduce a new function w to
=
u(q+1)/2;
then (3.7) leads (3.8)
~
C (1 + IIwllI2(QT») . 3
Recall Lemma 3.2, which implies u constant C4 such that
E
£2 (QT); so that there exists a positive
IIwIIL4/(q+l)(QT) ~ C4 .
Let q* be any number satisfying q* > max{2,4/(q + I)}. Using Holder's inequality we see from the above estimate that
IlwIII2(QT) ~ Ilwll~~l"(~T)llwll~~/(q+l)(QT) ~ Cr'llwll~~l"(~T)'
(3.9)
where
(~_ q1* )
A=
/
(q: 1 _
q~ )
.
Here we also use Gagliardo-Nireberg's inequality; for q* E [~, ~~21 II W IILq" (n)
~ M* (11V'wll~2(n) Ilwll~-;-:/(q+l)(n) + Ilwll£l(n») ,
(3.10)
with a positive constant M* and B=
(~-~) / (~+~). 2q q* N 2q
Setting w = w(t) in (3.10) and integrating the resulting expression with respect to t one can prove (3.11)
~C
5 (
sup
09:::;T
IIw(t)II(G~~~~l)(n) Jo(T IIV'w(t)llf2~n) dt + 1) ,
with C 5 > O. Choose q* such that q*B = 2; that is, q* = 2+4q/{(q+ l)N}. Recalling the definition of ET in (3.8) we see from (3.11) q" II w Il Lq"(QT)
(N+2)/N) +1 .
~ C5 ET
(3.12)
Then it follows from (3.8), (3.9) and (3.12) that
ET
~ C3 (1 + Crllwll~~:(~~») ~C6 (I + EfJ,)
(3.13)
291
with a positive constant C6 and f.L=
2(1 - >..)(N + 2)
N q*
<1.
Thus (3.13) implies sup IIu(t)II1q(f1)
O:St:ST
= sup Ilw(t)II~%~i~12)(f1) ~ ET ~ C7 O:St:ST
with some C7 > o. Finally, estimate for IIV'ull£2(QT) comes from (3.7) by setting q
=
1. 0
Remark 3.1. In case c5 > 0, Proposition 3.1 has been extended for 4(N + 1) , q < N _ 2 by an elaborate work of Tu<;>c [23, Lemma 2.2.5]. 4. Proofs of theorems
In this section we will give ideas of proofs for global existence results stated in §2. 1. Idea of proof of Theorem 2.3 Let T be any positive number. We will briefly explain the procedure to derive the Holder continuity of Ut, ~u, Vt and ~v as well as u, v, V'u and V'v. (i) Holder continuity of v and V'v. Recalling Proposition 3.1 we see from maximal regularity theory for the second equation in (P) (see, e.g., (3.5)) that v E W,;,l(QT), which together with the embedding theorem (see [8, Lemma 3.3, p. 80]) assures the Holder continuity of v and V'v with respect to (x, t) E QT. (ii) Loo estimate of u. The idea to show the above assertion is to rewrite (3.1) in the following divergence form: Ut
=
a) Na LNax.a ( a(x, t) a:. +L ax. (ai(x, t)u) + b(x, t)u, i=l'
•
i=l
•
where
a(x, t)
av
= d1 (1 + av(x, t) + 2,u(x, t)), ai(x, t) = d1 a-a (x, t) Xi
and
b(x, t) = a(1 - u(x, t) - cv(x, t)).
(4.1)
292
Since U can be regarded as a weak solution of (4.1), one can apply the maximum principle in [8, p. 181] to see V>O(QT) boundedness of u. (iii) Holder continuity of u. By (i) and (ii), a,ai and b appearing in (4.1) are bounded functions. Therefore, it follows from the regularity theory for a weak solution of (4.1) (see, e.g., [8, Theorem 10.1, p. 204]) that u is Holder continuous for (x, t) E
QT· (iv) Holder continuity of Vt and ~v. We go back to (3.2), where f = bv(1- du - v) is Holder continuous with respect x, t by (i) and (iii). Hence the famous Schauder estimate implies the Holder continuity of Vt and ~v for (x, t) E QT. (v) Holder continuity of Ut and ~u. It is convenient to introduce a new function satisfies, by virtue of (3.1),
Wl =
(1+av+,u)u, which (4.2)
with h(x, t) = au(1- u - cv)(1 + av + 2,u) + aUVt. Note that 1 + av + ,U and h are Holder continuous with respect to (x, t) E QT by (iii) and (iv). Application of Schauder estimate to (4.2) implies Wl E C2+8,(2+8)/2(QT) with a positive constant () > O. This fact also assures that Ut, \7u and ~u are Holder continuous for (x, t) E QT. II. Outline of proofs of Theorems 2.4 and 2.5
We will briefly state the outline of the proof of Theorem 2.4. From (P), v can be regarded as a weak solution of
Vt
=
\7.
«1 + 2<5v)\7v) + bv(1 -
du - v);
so that it follows from the Holder continuity result [8, Theorem10.1, p. 204] that v E Cp,p/2 (QT) with some p > o. Let W2 = (1 +<5v)v be a new function. Then it satisfies
W2,t = d 2(1
+ 2<5V)~W2 + g(x, t)
(4.3)
with 9 = bv(1 + 2<5v)(1 - du - v) E Lq(QT) for any 1 < q < 2(N + 1)/(N - 2) by Proposition 3.1. Since the coefficient in (4.3) satisfies d2(1 + 2<5v) E Cp,p/2(QT) and 9 E Lq(QT), the parabolic regularity theory (see, e.g., [8, Theorem 9.1, pp. 341-342]) implies W2 E W,i,1(QT) for q < 2(N + 1)/(N - 2). Hence the embedding result [8, Lemma 3.3, p. 80] gives \7w2 E L(N+2)q/(N-2- q)(QT), which implies \7v = \7w2/(1 + 2<5v) E
293 for q < 2(N + l)/(N - 2). These facts verify the assumptions of the maximum principle in [8, Theorem 7.1, p. 181] and enable us to apply this principle to (4.1) if q satisfies L(N+2)q/(N-2- q )(Qr)
N +2 2(N + 1) -2-
(4.4)
Here we should note that it is possible to choose q satisfying (4.4) if and only if N = 2,3,4,5. Application of the maximum principle to (4.1) yields
u E £<>O(Qr). For the rest of proof, see the arguments developed in [3,4]. The idea of the proof of Theorem 2.5 is essentially the same as that of Theorem 2.4. In view of Remark 3.1, it is possible to use the following relations
N +2 4(N + 1) -2-
III. Idea of proof of Theorem 2.6 We will give another approach which uses the maximum principle to a new function and does not need LP(n)-estimates. This kind of approach has been proposed by D. Le in [9] (see also a series of works [10-12]). Here we will follow the idea used by Tui?c [22]; that is, in a sense, refinement of the proof of Le and Nguyen [12]. The first step is to rewrite (P) in the following form Ut Vt
= V· [P(u, v)Vu + R(u)Vv] + au(l- u = V· [Q(u, v)Vv] + bv(l - du - v)
ou=ov=o
ov
u( . ,0)
ov
= Uo
~
0, v(., 0)
= Va
~
0
cv)
in
nx
(0,00),
in
nx
(0,00),
on
on x
in
n,
(0,00),
(4.5)
294
where P(u,v)=d 1(1+av+2')'u),
Q(u,v)=d 2(1+26v),
R(u)=d1au.
Set P - Q
d*
f(u,v)=~=
+ a*u -
b*v
u
with b* = 2d26 - d 1a . a d1a Let m be the positive number defined in Lemma 3.1. We denote r {(u,v) E R2; u:::O: O,m:::O: v:::O: O} and rK = {(u,v) E R2; u > K,m:::O: v:::O: O} for any positive number K. We choose a sufficiently large number Ko such that *
2')'
a =-,
d*+a*u-b*v > 0
b*[d*(b*-1)+a*b*u-b*(b*-1)v] > 0
and
for 0 :s: v :s: m and u :::0: K o/2. The next step is to define G(u, v) = 'ljJ(u)g(u,v) for (u,v) E
b*[d*(b*-l)+a~b*u-b*(b*-l)vl b*
(b*-l) log
(
)
g(u, v) = v-d* * --+a logu
if b*
r,
-I- 0
where
if b* 10,1, if b*=l, (4.6)
u
if b* =0,
v+d* logu+a*u
and'ljJ is a smooth function such that 0 :s: 'ljJ(u) :s: 1, 'ljJ(u) = 1 for u :::0: Ko and 'ljJ(u) = 0 for u :s: K o/2. We should note that G is a smooth function satisfying Gu(u,v)
=
f(u,v)Gv(u,v) :::0: 0
in
rKo
and
lim G(u, v)
u-->oo
=
00.
We are ready to apply the maximum principle. The strategy is to introduce a new function w(x, t)
= G(u(x, t), v(x, t».
Then it is possible to see that, for all (x, t) satisfying u(x, t) > K o , Wt
= Ptlw
+ h(x, t) + ¢(x, t)
with h(x, t)
=
(2')'G u - PG uu )[V'u[2
+ 2(aGu -
PGuv)V'u· V'v
+(26Gv - PG vv )[V'v[2, ¢(x, t) = au(1 - u - cv)G u
+ bv(1 -
du - v)G v .
(4.7)
295 Under the assumptions of Theorem 2.6, one can show hex, t) ::; 0, ¢;(x, t) ::;
°for any (x, t) whenever u(x, t) is sufficiently large. Therefore, the maxi-
mum principle is applicable to (4.7); the contradiction argument enables us to conclude that w is bounded. This fact also implies that u is bounded. The rest of proof is essentially the same as Theorem 2.4.
IV. Idea of proof of Theorem 2.7 The idea to prove Theorem 2.7 is quite similar to that of Theorem 2.6. We will follow the arguments of Le [10]. Rewrite (1.1) in the following form ut {
Vt
+ au(l- u = \7. [Q(u, v)\7v + S(v)\7u] + bv(l- du =
\7 . [P(u, v)\7u + R(u)\7v]
cv)
in
nx
(0,00),
v)
in
nx
(0,00),
with zero Neumann boundary conditions and initial conditions. Here P(u, v) R(u)
= d l (1 + av + 21'U) , Q(u, v) = d2 (1 + (3u + 2<5v) ,
= dIau,
S(v)
Let (u, v) be a solution of (1.1) on
nx
= d 2 (3v.
[0, T]. Set
r* = {(u(x, t), vex, t)) E R2; (x, t) En x [0, T]}. As in [10]' assume that there exists a C2 function H(u, v), which is defined in a neighborhood ro of r*, with the following conditions. (H.O)
There exists a constant Ko such that (HuF
+ HvG)(H -
K)+ ::;
° for all
(u, v) E
ro
and
K:2: Ko
where F(u, v) = au(l - u - cv), G(u, v) = bv(l - du - v) and w+ = max{w,O}. (H.1) There exists a positive constant Al such that {Hu (P\7u (P\7u
+ R\7v) + Hv(Q\7v + S\7u)}· \7H:2: AII\7HI2,
+ R\7v)· \7Hu + (Q\7v + S\7u)· \7Hv :2:
°
for (u,v) E r* n {(u,v) E R2; H(u,v) :2: Ko}. Here \7 denote the gradient with respect to x-variable. (H.2) H(u, v) ~ 00 if (u, v) ~ 00.
296
Under these assumptions, it can be shown that
k
(H(u(x, t), v(x, t)) S;
k
k
K)~ dx + Al fat ds IV'xH(u(x, s), v(x, s)Wdx
(H(uo(x),vo(x)) -
K)~
dx
for any K ~ Ko. Therefore, if we take a sufficiently large number K > Ko satisfying K ~ maxxEo{H(uo(x), vo(x))}, then we see from the above inequality
H(u(x, t), v(x, t)) S; K
u(x, t) En x (0, T).
for
This estimate,
together with (H.2), implies the boundedness of (u(x, t), v(x, t)) for (x, t) En x (0, T). The essential point in the proof of Theorem 2.7 is to choose H(u, v) satisfying (H.O), (H.1) and (H.3). The assumptions on diffusion coefficients are sufficient conditions for such H(u, v) to exist. For details, see the work of Le [10]. 5. Concluding remarks
A. In Theorems 2.4,2.5 and 2.6, we have imposed restrictions on the space dimension or diffusion coefficients for the global existence of solutions for (P) in case 8 > O. It is an open problem to establish the global existence result for (P) without any restrictions. B. Let N ~ 3. In each proof of Theorems 2.3-2.6, the positivity of self diffusion coefficient , plays a crucial role. Although the global result is valid in case N = 2 (see [14]), it is a difficult problem to show the global existence of solutions to (P) in case N ~ 3 and a > 0, , = 0 even if 8 = 0 in the second equation. Consider the following related problem; Ut =
dI~[(l
Vt =
d2~V
+ av)u] + au(l - u Pl
+ bv(l - du P2
-
-
v)
ou=ov=O
ov ov
u( . ,0) = Uo
~
0, v(·, 0) = Vo ~ 0
cv)
in
nx
(0,00),
in
nx
(0,00),
on
an x (0,00),
in
n,
(5.1)
where d I , d2, a, b, c, d, a, PI, P2 are positive constants. This system were studied in [17], [19] and [28]. In particular, Redlinger [19] and the author [28] have shown that (5.1) possesses a unique global solution (u, v) if PI > P2.
297
The last condition excludes our problem (P). So it will be interesting if we can show that (P) admits a global solution in case 'Y = O.
C. Theorems in §2 give no information on the asymptotic behavior of solutions of (P) as t -4 00. There are only a few works on the dynamical theory associated with (P). If the space dimension N is one and some assumptions are imposed on coefficients of (P), Shim [21] has shown the uniform boundedness of solutions to (P) and their convergence to a constant steady state as t -4 00. When N is two, global existence results ofYagi [26,27] and LouNi-Wu [14] have been extended by Le [9]. He has established the uniform boundedness of solutions of (P) and shown the existence of a global attractor. See also [10, 12], where the uniform boundedness of solutions with respect to some Holder norms is proved. The study of dynamical theory for (P) is a very important open problem. As for the stationary problem associated with (P), see, e.g., a survey article by Ni [15] in case of zero Neumann boundary conditions and one by the author [29] in case of zero Dirichlet boundary conditions.
References 1. H. Amann, Dynamic theory of quasilinear parabolic equations II. Reactiondiffusion systems, Differential Integral Equations, 3 (1990), 13-75. 2. H. Amann, Linear and Quasilinear Parabolic Problems, Vol. 1 Abstract Linear Theory, Birkhauser Verlag, Basel-Boston-Berlin, 1995. 3. Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion, Discrete Contino Dynam. Systems, 9 (2003), 1193-1200. 4. Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contino Dynam. Systems, 10 (2004), 719-730. 5. P. Deuring, An initial-boundary-value problem for a certain densitydependent diffusion system, Math. Z., 194 (1987), 375-396. 6. T. Ichikawa and Y. Yamada, Some remarks on global solutions to quasilinear parabolic system with cross-diffusion, Funkcial. Ekvac., 43 (2000), 285-301. 7. J. U. Kim, Smooth solutions to a quasilinear system of diffusion equations for a certain population model, Nonlinear Anal. 8 (1984), 1121-1144. 8. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, Rhode Island, 1988. 9. D. Le, Cross-diffusion systems on n spatial dimensional domains, Indiana Univ. Math. J., 51 (2002), 625-643.
298 10. D. Le, Global existence for a class of strongly coupled parabolic systems, Ann. Mat. Pum Appl. (4),185 (2006), 133-154. 11. D. Le, L. V. Nguyen, T. T. Nguyen, Shigesada-Kawasaki-Temmoto model on higher dimensional domains, Electron J. Differential Equations 2003, No. 72, 12pp. 12. D. Le and T. T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension, Proc. Amer. Math. Soc., 133 (2005), 1985-1992. 13. Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains, Discrete Contino Dynam. Systems, 12 (2005), 185-192. 14. Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contino Dynam. Systems, 4 (1998), 193-203. 15. W.-M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 1, Edited by M. Chipot and P. Quittner, pp. 157-233, Elsevier, Amsterdam 2004. 16. A. Okubo, L. A. Levin, Diffusion and Ecological Problems: Modern Perspective, Second edition, Interdisciplinary Applied Mathematics, 14, SpringerVerlag, New York, 2001. 17. M. A. Pozio and A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system, Nonlinear Anal., 14 (1990), 657-689. 18. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. 19. R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics, J. Differential Equations, 118 (1995), 219-252. 20. N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. 21. S. A. Shim, Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion, J. Math, Anal. Appl., 279 (2003), 1-21. 22. P. V. TU9C, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions, Proc. Amer. Math. Soc., 135 (2007), 3933-3941. 23. P. V. TU9C, On global existence of solutions to a cross-diffusion system, Ph.D dissertation, University of Minnesota, 2007. 24. Y. Wang, The global existence of solutions for a cross-diffusion system, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), 519-528. 25. A. Vagi, Some quasilinear parabolic problems in applied mathematics, Proceedings of Bologna Conference on Functional Analysis and Related Topics, pp. 255-267, Lecture Notes in Math., 148, Marcel Dekker, 1993. 26. A. Vagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Anal., 21 (1993), 603-630. 27. A. Vagi, A priori estimates for some quasi linear parabolic system in population dynamics, Kobe J. Math., 14 (1997), 91-108.
299 28. Y. Yamada, Global solutions for quasilinear parabolic systems with crossdiffusion effects, Nonlinear Anal., 24 (1995), 1395-1412. 29. Y. Yamada, Positive solutions for Lotka-Volterra systems with crossdiffusion, Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 6, Edited by M. Chipot, pp. 411-501, Elsevier, Amsterdam, 2008.
300
DYNAMICS OF GLOBAL SOLUTIONS OF A SEMILINEAR PARABOLIC EQUATION Eiji Yanagida·
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan • E-mail: [email protected] This article is concerned with the behavior of global solutions of a parabolic partial differential equation with a power nonlinearity. Depending on parameter values and initial data, various behavior of global solutions can be observed such as grow-up, convergence, quasi-convergence and non-convergence. We give precise descriptions of the relationship between the spatial decay rate of initial data and the large-time behavior of solutions.
Keywords: Semilinear parabolic equation; global solutions; attractivity; convergence; grow-up; self-similar solutions.
1. Introduction
The purpose of this article is to survey recent progress on the behavior of global solutions of a semilinear parabolic equation with a power nonlinearity. Results described below are mostly obtained through a joint research project with Marek Fila, John R. King, Peter Polacik and Michael Winkler. Consider the Cauchy problem Ut=b.U+UV , {
u(x,O)
xE]RN,
= uo(x), x
t>O,
(1)
E ]RN,
where b. is the Laplace operator with respect to x. Some of the results below can be extended to sign-changing solutions, but for simplicity, we assume that the initial value Uo is a nonnegative continuous function on ]RN so that the solution remains nonnegative for all t > O. This simplelooking problem is known to have very rich mathematical structure and has attracted much attention (see a recently published book of QuittnerSouplet46 and the references cited therein). Also, as demonstrated at many occasions, by studying (1) one may gain deeper insights into the dynamics of more general parabolic problems on ]RN.
301
The aim of this paper is to discuss the dynamics of global solutions of (1). It turns out that depending on parameter values and initial data, various behavior of global solutions can be observed such as grow-up, convergence, quasi-convergence and non-convergence. In certain situation, it can be shown that the large-time behavior of solutions is determined by the decay rate of initial data at the spatial infinity. Moreover, we can give precise descriptions of the relationship between the spatial decay rate of initial data and the large-time behavior of solutions. This article is organized as follows. In Section 2, we summarize known results of positive steady states, and introduce a few critical exponents concerning the existence of regular and singular steady states. We also present some static properties of steady states, and define some important numbers that play an important role in the subsequent sections. In Section 3, we give a fundamental result concerning the global stability of steady states. We also consider two general solutions that approach each other, and discuss the rate of approach. In Section 4, we consider the existence of nonconvergent solutions. It is shown that if initial data decay in an irregular way as Ixl ~ 00, then the solution may behave in a very complicated way. In some cases, the solution does not stabilize but approaches a set of steady state (not a single one). To describe this process, we introduce the notion of quasi-convergence, and show that the solution enjoys this property. In Section 5, we show that if initial data decay more slowly than any other regular steady states but below the singular steady state, then the solution becomes unbounded as t ~ 00. Given a more specific decay rate of initial data, we can determine the relation between the the decay rate of initial data and the grow-up rate of solutions. In Section 6, we consider the behavior of solutions that converge to the trivial solution. We classify the solutions into three classes; slowly decaying solutions, solutions with a critical decay rate, and rapidly decaying solutions. In Section 7, we introduce forward self-similar solutions, and consider solutions that decay like self-similar solutions. In particular, we discuss the convergence rate of solutions of (1) to a self-similar solution. In Section 8, we consider the behavior of rapidly decaying solutions. If a solution decays to 0 rapidly, then the nonlinear term does not play an important role, and the solution is shown to behave like a solution of the linear heat equation. We also determine the convergence rate to a self-similar solution of the linear heat equation.
302
2. Properties of steady states
2.1. Critical exponents It is known that the structure of solutions of (1) drastically changes when the exponent P exceeds certain special values called critical exponents. In this subsection, we collect some critical exponents that are important in this article. First of all, the Fujita exponent defined by
2 PF:= 1 + N borders the case of the existence of positive global solutions (Fujita 16 ). Namely, there is no positive global solution if 1 < P ~ PF, while there exist some positive global solutions if P > PF. See Levine30 and Deng-Levine4 for generalizations and variations of this result. The Fujita exponent is also related to the existence of positive forward self-similar solutions (see Section 7). The exponent if N > 2, if N
~
2,
appears as the borderline for the existence of a singular solution explicitly given by
IPoo(x) := L[x[-m, where m ad L
= L(p, N) are defined by 2
m:=--, p-1
L:= {m(N - 2 - m)}l/(P-l).
The function IPoo is defined for P > Psg, and it is a weak solution of (1). Note that L > 0 for P > Psg and L --+ 0 as P 1 Psg. The Sobolev exponent if N > 2, if N
~
2,
is known to be critical for the existence of positive steady states of (1). Namely, there exist classical positive solutions u = u(x) E Co(]RN) of ~u+uP
= 0 on ]RN,
303 if and only if P ~ PS. For P ~ Ps, there is a one-parameter family of positive radial steady states u = 'Pa(x), where 'Pa(x) = l1>(r), r = lxi, is the unique solution of I1>rr {
+N;
1 I1>r
+ I1>P = 0,
r > 0,
11>(0) = a > 0.
We note that for each a > 0, l1>(r) is strictly decreasing in r and satisfies l1>(r) ~ as r ~ 00. By the rescaling invariance of (1), 'Pa satisfies
°
(2) where 'Pi is the (unique) radial steady state with a = 1. It is clear from the analysis of second order ODEs that {'Pa, a E lR} contains all radial steady states of (1). However, the question whether all positive steady states have to be radial about some point does not seem to have been answered completely. See Gidas-Ni-Nirenberg 18 and ZOU 51 for symmetry results under additional conditions. The Joseph-Lundgren exponent (Joseph-Lundgren26)
Pc :=
{
(N _2)2 -4N +8~ (N - 2)(N - 10)
if N
00
if N ::; 10,
> 10,
which is often written in the form N - 2JN -1
Pc -- N - 4 - 2JN -1
if N > 10,
has appeared in several different studies of (1) and related problems (DoldGalaktionov-Lacey-V azquez,6 Galaktionov-Vazquez, 17 Gui-Ni-Wnag, 19,20 Herrero-Velazquez 22 and Wang 49 ). In particular, it was shown by Wang 49 that if P ~ Pc, then 'Pa is strictly increasing in a for each x. Furthermore, for each x one has lim 'Pa(x) = 0,
a-->O
It is known that if PS ::; P < Pc, then the graph of any steady state intersects the graphs of all other regular steady state and the singular steady state 'Poo. We will see that such structural change of steady states leads to the stability change of steady state.
304
2.2. Asymptotic behavior of steady states
To formulate our results, we need further properties of the radial steady states of (1) concerning the asymptotic behavior of 'Po as Ixl --> 00. For p > Pc, the regular radial steady state 'Po has the following expansion as
Ixl --> 00
(Li 31 ):
'Po(x)
=
Llxl- m
-
a(a)lxl- m -
A1
+ o(lxl- m - A1 )
as
Ixl --> 00,
(3)
where Al is a positive constant given by
Al :=
N - 2 - 2m - V(N - 2 - 2m)2 - 8(N - 2 - m) 2 .
and a(a) is a positive number depending on a. Observe in particular that all positive steady states have the common leading term in the expansion at Ixl = 00, and they only differ in the second term. By (2) and (3), we have
so that a(a) = a- AI/ m a(l). Hence a(a) is monotone decreasing in a and ranges over (0,00) as a varies on (0,00). We note that Al is the smaller root of the quadratic equation A2 - (N - 2 - 2m)A + 2(N - 2 - m)
= 0.
(4)
This equation has two positive roots if and only if P > Pc, and the larger root is given by A2 :=
N - 2 - 2m + V(N - 2 - 2m)2 - 8(N - 2 - m) . 2
The roots of (4) play an important role in the subsequent sections. In the critical case p = Pc, (4) has a double root
Ao:=
N -2 -2m 2 '
and 'Po is expanded as
'Po(x) = Llxl- m
-
alxl- m -
AQ
In Ixl
+ h.o.t.
as
Ixl --> 00.
(5)
The logarithmic term in the expansion sometimes causes technical difficulties.
305
3. Attractivity properties of steady states
3.1. Stability of steady states In the following, we often assume the following condition on initial data:
(At) P > Psg, and O:S uo(x) :S 'Poo(x) for all x E ]RN
\
{O}.
Under this condition, the solution of (1) exists globally in time, and remains below the singular steady state for all t > 0 (see, e.g., Polacik-Yanagida39 ). If Ps :S P < Pc, any two distinct positive steady states intersect, and this fact leads to unstable behavior of nearby solutions. Indeed, for Ps :S P < Pc any positive steady states 'Po is unstable in any reasonable sense, and moreover, for each Uo :::: 'Po, Uo ¢ 'Po, the solution of (1) blows up in finite time. Conversely, when the positive steady states are ordered, it is easy to see from a comparison theorem that each steady state is stable under some perturbations. A more precise local stability analysis was done by Gui-Ni-Wang. 19,20 They showed that for P :::: Pc, every positive steady state is locally stable with respect to certain weighted Loo norms. Building on the results of Gui-Ni-Wang, 19,20 we can extend the local stability to global attractivity properties of steady states (and also of general solutions) in a class of admissible functions. We will give a sharp condition on the initial function Uo for the solution of (1) to converge to a given steady state 'Po. This renders a partial description of the domain of attraction of 'Po. The following theorem shows that if the initial value Uo decays like 'Po up to the second order, then the solution of (1) converges to 'Po. In other words, any deviation from a steady state with sufficiently fast decay will vanish as t -- 00.
Theorem 3.1 (Pohicik-Yanagida39 ). Let P > Pc. Assume that Uo satisfies (AJ) and lim Ixlm+Alluo(x) - 'Po (x)
Ixl-+oo
I = O.
(6)
Then the solution u of (J) satisfies
lIu(" t) - 'Po 0
II Loo (lRN)
--
0
as t --
00.
In the critical case P = Pc, this theorem holds true if (6) is replaced by lim Ixlm+>'O(ln Ixl)-lluo(x) - 'Po(x)1 = 0
Ixl-+oo
(cf. (3) and (5». Since Uo is not assumed to be close to 'Po for finite x, Theorem 3.1 describes a rather large domain of attraction of the positive
306 steady state. We note that the condition (6) is sharp for a =/:- O. In fact, for arbitrarily small c, we can find an initial value such that lim sup Ixlm+>'lluo(x) - 'Pa(x)1 = c > 0, Ixl->oo liminf Ixlm+'xlluo(x) - 'Pa(x)1 = 0, Ixl->oo for which the conclusion of Theorem 3.1 fails. For the trivial solution 'Po == 0, we can show a stronger stability result (see Theorem 6.1). A proof of Theorem 3.1 is obtained by the comparison technique. A key idea is to use a positive solution of the linearized equation
x
E ]RN, t
> 0,
(7)
with sufficiently large (3 (> a). By the linear stability result of Gui-NiWang,19 any bounded solution of (7) satisfying a certain decay condition converges to 0 as t ---- 00. Using this fact and 'Pf3 > 'Pa for all x, we can show that
u+(x, t) := min{ 'Poo(x), 'Pa(x)
+ vex, tn,
u-(x, t) := max{O, 'Pa(x) - vex, tn, are a supersolution and a subsolution of (1), respectively, that converge to 'Pa as t ---- 00. Then Theorem 3.1 follows from the comparison principle.
3.2. Approach of two solutions Theorem 3.1 can be extended to positive solutions that are not necessarily steady states. Let u and U denote solutions of (1) with initial data Uo and uo, respectively. Roughly speaking, if two initial data are bounded by the singular steady state and are sufficiently close to each other as Ixl ---- 00, then the solutions of u and u approach each other. This implies that the large-time behavior of solutions of (1) is determined only by the behavior of initial data as Ixl ---- 00 and the initial data for finite x do not play any role.
Theorem 3.2 (PoIM:ik-Yanagida39 ). Let p > Pc. Suppose that Uo and Uo satisfy
o :s; uo, Uo :s; 'Pf3(x), with some 0 < (3 <
00,
x
E]RN
(8)
and
lim [x[m+>'lluo(x) - uo(x)1 = O. Ixl->oo
(9)
307
Then the corresponding solutions u and [[u(·, t) - u(·, t)[[u"'(IRN)
u of (1) satisfy -->
0
as t
--> 00.
Again this theorem holds true for p = Pc if (9) is replaced by lim Ixlm+Ao(ln Ix[)-lluo(x) - u(x)1 = Ixl--+oo
o.
Needless to say, if Uo = <Pa, then Theorem 3.2 reduces to Theorem 3.1. In particular, if Uo - Uo has a compact support, then the difference u - u tends to 0 as t --> 00.
3.3. Rate of approach Given a specific behavior of Uo - Uo as Ixl --> 00, we can compute how fast these two solutions approach each other. In particular, in the case of Uo = <Pa(x), the rate of approach corresponds to the convergence rate to the stationary solution. The following result was proved by Fila-Winkler-Yanagida12 for m + 23 >'1 < I < m + >'2 and later by Hoshino-Yanagida in a wider range of >..
Theorem 3.3 (Fila-Winkler-Yanagida, 12 Hoshino-Yanagida23 ). Let P > Pc. Suppose that Uo and Uo satisfy (8) and
[uo - uol S c(l with some c > 0 and I > m
(i) If m
+ >'1 < l <
m
+ [x[)-l, x
E
]RN
+ >'1.
+ >'2 + 2,
then there exists a constant C
> 0 such
that
lIu(·, t) - u(·, t)IILOO(IRN) S C(l
+ t)-(I-m->'!l/2
for all t > o. (ii) If I ::::: m + >'2 + 2, then for any small c > 0 there exists a constant Cc; > 0 such that
lIu(·, t) - u(·, t) IILoo (IRN) S Cc;(1 for all t >
+ t)-(A2- Al+ 2 )/2 +c;
o.
The next theorem shows that if luo - uol decays much faster in space, then we have a slightly better estimate than in Theorem 3.3 (ii).
308 Theorem 3.4 (Hoshino-Yanagida23 ). Let p > Pc. Suppose that Uo and Uo satisfy (8) and
with some constants c > 0 and v > such that
for all t >
o.
Then there exists a constant G > 0
o.
Theorem 3.3 (i) is no longer valid for l > m + >'2 + 2. In fact, a sharp universal lower bound, which applies to any initial data, is obtained by Fila-Winkler-Yanagida. 12
Theorem 3.5 (Fila-Winkler-Yanagida 12 ). Let p > Pc. Suppose that Uo and Uo satisfy (A 1) and
o.
Then for any
€
x E]RN
> 0 there exists a constant Ge > 0 such
Ilu(., t) - u(·, t)llu>o(lRN) ?: Ge(l + t)-(A2- Al+ 2)/2-e for all t >
o.
This lower bound implies that the convergence rate of Theorem 3.3 (i) can not be extended to the range l > m >'2 + 2. Proofs of the above theorems are obtained by a comparison technique that is based on matched asymptotic expansion. This expansion consists of two parts which are called the inner expansion and the outer expansion. The inner expansion is used to approximate the behavior of solutions near the origin, while the outer expansion is used to approximate the behavior near the spatial infinity. A key to the proof is the precise description of the outer expansion; we use a solution of the linearized equation at
+
3.4. Anisotropically decaying solutions In this subsection we consider the case when p > Pc and Ixl m +A1 (Llxl- m uo(x)) has a limit along any ray in ]RN emanating from the origin, but the
309 limit may vary with the direction. More specifically, we assume that Uo satisfies
where w = xllxl and b is a positive continuous function of wE SN-l. In this case, Theorem 3.1 can not be applied, but it can be shown that the solution of (1) converges to a steady state 'Po:. Moreover, rather surprisingly in the context of nonlinear equations, the limit equilibrium is explicitly determined by an average formula. . Here is the precise formulation of the result. Theorem 3.6 (Pohicik-Yanagida 42 ). Let P > Pc. Assume that Uo satisfies (Al) and (10) holds uniformly in wE SN-l. Then the solution of (1) satisfies
where
0:
is uniquely determined by
a(o:) =
IS~-ll1sN-l b(w) daw •
The proof of this theorem can roughly be outlined as follows. Using Theorem 3.2, we first observe that it is sufficient to prove the result for a particular solution u whose initial value Uo satisfies (10). To find such a particular solution, we show that the linearization of (1) around the singular steady state 'Poo has a self-similar solution that satisfies specific asymptotics as Ixl -.0 and Ixl -. 00. Then we are able to understand the behavior of u and thus prove the convergence result of Theorem 3.6. 4. Non-convergence and quasi-convergence
4.1. Non-stabilizing solutions When Uo behaves in an irregular way as Ixl -. 00, a rather complicated temporal behavior may be exhibited by the corresponding solution. Our result reveals an interesting and perhaps surprising possibility: the solution may approach continuum of steady states, not settling down to any particular one of them. Assume that (11)
310
for some 0 < 8 of (1) satisfies
< f3 < 00. Then, by the maximum principle, the solution u 'P8(X) ~ u(x, t) ~ 'P{3(x),
xE
]RN
for all t > O. Since 'P8, 'P{3 decay to 0 as Ixl ---. 00, the parabolic regularity implies that the set {u(·, t) : t ~ O} is relatively compact in Co(]RN). Hence by standard results, as t ---. 00, u(·, t) approaches its w-limit set
w(u):= {¢: u(·,t n ) ---. ¢ for some sequence tn ---. oo}. The set w(u) is nonempty, compact and connected in Co(]RN). An analogy with semilinear heat equations on bounded domains would suggest that for any initial condition, the solution converges to a steady state. However, it turns out that the solutions may fail to stabilize. We can find a solution that oscillates between two equilibria as t ---. 00 and its w-limit set coincides with the arc of equilibria joining the two:
Theorem 4.1 (Pohicik-Yanagida39 ). Let P > Pc. Given 0 < 8 < f3 < 00, there exists a continuous function Uo satisfying (11) such that w(uo) = {'P-y: 8 ~ 'Y ~ f3}. We remark that nonstabilizing bounded solutions are not so difficult to encounter on unbounded domains. Even the linear heat equation on ]R possesses such solutions (Collet-Eckman3 ). An interesting point about our example is that the solution decays to 0 at spatial infinity (uniformly with respect to t) and we can even take the initial data arbitrarily small in the supremum norm. In order to prove Theorem 4.1, we first take a sequence {o:d that is dense and contained in [8, f3J. Then we take initial data such that
uo(x) == 'P<>i(X)
for ai ~ Ixl ~ bi ,
where {ad an {b i } are sequences that satisfy
o < al < b1 < a2 < b2 < ... < ai < bi < ... and are sufficiently separated. Taking such sequences appropriately, and using the global attractivity of steady states and the continuous dependence of solutions with respect to initial data, we can show that for every 0: E (8, (3), there is a sequence {tj} such that tj ---. 00 and Ilu(·, tj) - 'Pa(·) IILoo(JRN) ---. 0 as j ---. 00. (See Theorem 4.4 for a more general result.) Using the initial data given as above, we can show
311
The converse inclusion
w( uo)
C
{ip-y : 8 :::; 'Y :::; ,B}
is also proved in Pohicik-Yanagida39 under some extra condition on initial data. A more general result is given in the next subsection. 4.2. Quasi-convergence
We address a natural complementary question whether for each Uo satisfying (11) the solution of (1) approaches a set of steady states (henceforth such solutions are referred to as quasi-convergent). The question is not trivial as there is no obvious Lyapunov functional for (1) defined along all solutions considered here, not even along differences of such solutions. Other methods, such as sweeping-type arguments that usually work well in the presence of a simply ordered curve of equilibria for equations on bounded domains, do not apply here either. By such methods one typically proves the convergence to a single equilibrium (see e.g. Smith44 ), which is not always the case here. We give a positive answer to the question for P > Pc.
Theorem 4.2 (Pohicik-Yanagida41 ). Let P isfy
for some
°<
ip,,(x) :::; uo(x) :::; ip{3(x), 8
< ,B :::; 00,
> Pc. x
E
If Uo E Co(JR N
)
sat-
(12)
JRN
then
w(uo)
C
{ip-y : 8 :::; 'Y :::; ,B}.
To describe our method of proof, we first note that w(uo) consists of elements of Co (JR N ) that have entire trajectories contained in w(uo). Specifically, for each Z E w(uo), there is a solution u(·, t) of (1) defined for all t E JR such that u(·,t) E w(uo) and u(o) = z (see, e.g., Henry21). Also, by the comparison principle, (12) implies ip" :::; u(-, t, uo) :::; ip{3 for all t ?:: 0, and hence
(13) whenever u(·, t) E w(uo). Thus the quasi-convergence result is obtained if the following Liouville-type property is established.
°
Theorem 4.3 (Pohicik -Yanagida 41 ). Let P > Pc. If < 8 < ,B :::; 00 and u(-, t) is a solution of (1) on (-00,0] satisfying (13) for all t < 0, then u(·, t) == ipo. for some a > 0.
312
In order to prove this theorem, we first reformulate the problem in selfsimilar variables. Setting
x y = ~T-t'
s=-log(T-t),
1
u(x, t) = (T - t)- p-l v(y, s),
(14)
where u is a solution of (1) on (-00, T] and T is a fixed real number, we obtain the following equation for v:
Observe that applying this change of variables to a radial steady state CPa,
o < a < 00, we obtain an s-dependent solution
The solutions 'l/Ja(Y, s) are actually time translates of 'l/Jl, as, by (2), we have
Clearly, 'l/Ja(Y,s) below,
-->
0 as s
-->
00 for each Y E
]RN.
Furthermore, as we show
where CPoo is the singular steady state of (1) defined in (14), and it is also a singular steady state of (15). Hence, in the self-similar variables, the family of steady states corresponds to a heterocIinic connection (at least in the pointwise sense) between the singular steady state and the trivial steady state. Moreover, the connection is monotone decreasing in s, because CPa is increasing in a. Now if u is a solution satisfying (13) for all t E ]R, then the corresponding function v satisfies
It follows that v is also a heterocIinic connection from CPoo to 0 and stays between CPoo and 0 (in the pointwise ordering) for all times s. On the other hand, we can-show that such a heteroclinic solution is unique, up to a time shift, hence v (', s) == 'l/Jl (-, S - so) for some so. In the original variables, this is equivalent to the Liouville property stated above.
313
4.3. Birth-and-death of peaks In previous subsections, we showed the existence of nonstabilizing solutions that are bounded by the singular steady state, and hence localized near the origin for all t. In this subsection, we construct a solution that undergoes a sort of birth-and-death process. It first develops a peak at some position; then the peak disappears and appears again at a different position. This process is repeated with the positions and heights of the peaks prescribed arbitrarily. The following result is concerning the existence of such solutions.
Theorem 4.4 (Pohicik-Yanagida40 ). Let P :::: Pc. For any infinite sequence {(ai'~i,Ci)} with ai > 0, ~i E]RN andci > 0; there exists Uo such that the solution of (1) satisfies the following: (a) u(x, t) exists globally in time and decays to 0 as Ixl ---- 0 for each t. (b) There exists an increasing sequence of positive numbers {Si} such that Ilu(·, sdIILOO(IR N ) < Ci· (c) There exists an increasing sequence of positive numbers {td with ti E (Si' SH1) such that Ilu(·, ti) -
We would like to put this theorem in contrast with some results on asymptotic symmetry of positive global solutions for nonlinear parabolic equations. Busca-Jendoubi-Polacik2 considered positive bounded solutions of the equation
Ut
= ~u
+ f(u)
on ]RN,
(16)
where f is a C 1 function satisfying f(O) = 0 and f'(0) < O. Under certain assumptions on the decay of u at spatial infinity (in particular, the uniform decay with respect to t is required), they proved that the solution must converge to a steady state that is radially symmetric with respect to some point. Notice that the solution obtained in Theorem 4.4 exhibits a very different behavior; in particular, as t ---- 00, it does not symmetrize about any fixed point in ]RN. The crucial difference between (1) and (16) is the stability of the trivial solution. The assumption f' (0) < 0 implies that solutions of (16) converge to 0, as soon as they become sufficiently small in the LOO(]RN) norm. On the other hand, while the trivial steady state of (1) is stable in some norms, it is unstable in Co(]RN), which makes the recovery of the solution in Theorem 4.4 possible.
314
From a different point of view, the solution in Theorem 4.4 can be viewed as symmetrizing. Indeed, it is close to a shift of a radially symmetric solution at each large t > 0 so that this weaker form of symmetrization occurs. It is an interesting question to ask whether such weak symmetrization occurs for all positive global bounded solutions.
5. Grow-up of solutions
5.1. A sufficient condition for grow-up An interesting consequence of Theorem 3.1 is the existence of global unbounded solutions. Such solutions do not seem to exist for P < Ps. Indeed, for P :S PF there are no positive global (bounded or unbounded) solutions at all, so the question is only of interest for larger p. For PF < P < Ps it seems that not only all positive global solutions must be bounded, they have to decay to 0 as t -) 00 (see Section 6 for more details). Without any additional requirements on uo, the questions whether global solutions may be unbounded is open for Ps :S P < Pc. The following result clarifies the situation for P > Pc.
Theorem 5.1 (Pohicik-Yanagida39 ). Let P > Pc. Suppose that Uo satisfies (A 1) and lim Ixlm+>'I(lPoo(X) - uo(x)) =
Ixl~oo
o.
(17)
Then the solution of (1) is global and lim lIu(·, t)IILOO(lRN) =
t~oo
We note that in the critical case P replacing the condition (17) by
=
00.
Pc, Theorem 5.1 holds true by
lim Ixlm+>'I (In(lxl))-lIIPoo(x) - uo(x)1
Ixl->oo
= O.
The idea of the proof is quite simple. Fix a > 0 arbitrarily and define Uo by uo(x)
= min{IPa(x),uo(x)}.
If Ixl is sufficiently large, then IPa < Uo by assumption on uo(x) and (3), so that Uo == IPa for all sufficiently large lxi, and luol :S IPoo, Uo :S Uo. Then by the comparison principle, the corresponding solutions satisfy u(·, t) 2: u(·, t) for all t. From Theorem 3.1, we see that for each x E ]RN, IPoo(x)
2: liminfu(x,t) 2: lim u(x,t) = IPa(x). t~oo
t~oo
315
Since this is valid for arbitrary a and
= 0 and lim sup Ilu(" t) IIvx> (JRN) = 00. t->oo
5.2. Grow-up rate Concerning the rate of grow-up, the following upper bound is given in FilaWinkler-Yanagida. 11
Theorem 5.2 (Fila-Winkler-Yanagida l l ). Letp > Pc. Suppose thatuo satisfies (A 1) and
uo(x) ~ Llxl- m
-
blxl- l
for
Ixl > R
with some constants l > m + Al and b, R > O. Then there exist positive constants C and T such that the solution of (1) satisfies lIu(·, t)lIvX>(JRN) ~ Ct
=(1-",-)'1 )
2"1
for all t > T. The upper bound in this theorem is not optimal for large l. In fact, it is shown in Fila-Winkler-Yanagida11 that there is a universal upper bound independent of initial data. A sharp universal upper bound is obtained by Mizoguchi,33 where a spectral property of the linearized operator at the singular steady state is essentially used.
Theorem 5.3 (Mizoguchi 33 ). Let p > Pc. For any initial data Uo satisfying (A 1), there exist positive constants C 1 and T such that the solution of {1} satisfies
Ilu(" t)llu"'(JRN) ~ CIt
"'(-'2 - >'1 +2)
q
25
for all t > T . Moreover, for some initial data Uo satisfying uo(x) ~ Llxl- m
-
be-lxI2/4
for
Ixl > R
(18)
316
with some b, R > 0, the solution of (1) satisfies
Iluh t)IILOO(IRN)
~
m(A2- A)+2)
C2t
2Al
with some C 2 > 0 for all t > O.
Concerning the lower bound, a partial result was obtained by FilaWinkler-Yanagida l l in the case of IE (m+>Q,m+A2], and later extended by Fila-King-Winkler-Yanagida8 to the case I E (m + AI, m + A2 + 2]. Theorem 5.4 (Fila-King-Winkler-Yanagida8 ). Let p > Pc. Suppose that Uo satisfies (A 1) and uo(x) ~
Llxr m with some I E (m + Al, m + A2 + 2)
blxr
l
for
Ixl >
°
and b > O. Then there exists a positive constant C such that the solution of (1) satisfies
u(O, t)
~
Ct
m(!-m-A)) 2"'1
for all t > O.
This implies that the upper bound obtained in Theorem 5.2 is optimal for I E (m+Abm+A2 +2]. The next result is obtained immediately from Theorem 5.2. Theorem 5.5 (Fila-Winkler-Yanagidal l ). Let p > Pc. Suppose that Uo satisfies (A 1) and uo(x) ~ Llxl- m - blxl- l for Ixl > 0 with some I ~ m + A2 + 2 and b > 0. Then for any small EO > 0, there exists a positive constant Cc; such that the solution of (1) satisfies
for all t > O.
Our next theorem slightly improves a result in Theorem 5.3. Theorem 5.6 (Fila-Winkler-Yanagidal l ). Let p > Pc. Suppose that Uo satisfies (A 1) and uo(x) 2: Llxl-m - be-vlxl2
for
Ixl
~1
with some positive constants b and v. Then there exists a positive constant C such that the solution of (1) satisfies
u(O, t)
~
ct
m(A2- A)+2) 2"'1
317
for all t
> O.
For the proof, based on a formal expansion of radial solutions, we construct an approximate solution. Matching the approximate solutions in the outer region and in the inner region, we can derive the expected grow-up rate. Modifying the approximate solution, we can construct an appropriate super- and subsolutions, which lead to the exact grow-up rate.
5.3. Grow-up in the critical case The rate of the grow-up in the case P = Pc is discussed in Fila-King-WinklerYanagida. 9 In this case, a logarithmic term appears in the precise grow-up rate. Theorem 5.7 (Fila-King-Winkler-Yanagida9 ). Let P that Uo satisfies (A 1) and Llxl- m
-
b_lxl- 1 ::; uo(x) ::; Llxl- m
-
b+ Ixl- l
=
Pc. Assume
for Ixl > R
with some l E (m + A, m + A + 2),0 < b+ < Land R > O. Then there exist positive constants c, C and T > 1 such that the solution of (1) satisfies ct for all t
m(l-m->.)
2X
m
(lnt)"X::; Ilu(·,t)IIL'X>(JRN) ::; Ct
m{l-m->')
2X
m
(lnt)"X
(19)
> T.
The proof is similar to the supercritical case P > Pc, but because of the logarithmic term in the expansion (8), the construction of suitable superand subsolutions becomes technically much more difficult. As in the supercritical case P > Pc, Theorem 5.7 can not be extended to l > m + A + 2. In fact, the next result gives a universal upper bound on solutions which shows that the lower bound in (19) cannot be extended to l > m+A+2. Theorem 5.8 (Fila-King-Winkler-Yanagida9 ). Let p = Pc. Assume that Uo satisfies (A1). Then for any c > 0, there exist positive constants Cc and T such that the solution of (1) satisfies Ilu(.,t)IILOO(JRN) ::; Cct T +C for all t > T.
The proof of Mizoguchi 33 on the universal upper bound (18) for the supercritical case p > Pc is not applicable to the critical case p = Pc, because the required spectral properties are only known to hold for p > Pc.
318
In Fila-King-Winkler-Yanagida,9 a different technique is used to derive the universal upper bound as above.
6. Convergence to the trivial steady state
6.1. Sufficient conditions for the converyence to zero In this section, we consider solutions that decay to 0 as t - t 00. For P > PF, various sufficient conditions have been obtained for the decay. See, e.g., Fujita,16 Lee and Ni,29 Galaktionov and Vazquez, 17 Kawanag0 28 and other references. 25 ,38,43,47 For PF < P < Ps it seems that any positive global solutions must decay to 0 as t - t 00. So far this result has been proved under extra conditions on P or uo. For example, it is known to be true for global solutions whose initial data Uo have fast decay at spatial infinity or at least are square integrable (see Kavian 27 and Souplet 45 ). Under another set of extra conditions, involving in particular the radial symmetry of uo, the result is proved by Matos-Souplet. 32 For Ps ~ p, the same result is not valid in general (steady states are bounded solutions that do not converge to zero), but Mizoguchi34 proved that it does hold provided the initial value of the solution is radially symmetric, has compact support and other technical conditions are satisfied. Here, in the super critical case P > Pc, we first derive a sharp sufficient condition on solutions to decay to 0 as t - t 00.
Theorem 6.1. Let P > Pc. Assume that Uo satisfies (A1) and lim Ixl m+>'l {
-t
0
as t
(20)
- t 00.
This improves the description of the domain of attraction of 0 given in Gui-Ni-Wang,20 where the condition sUPlxl>1Ixlm+cuo(x) < 00 for some € > 0 is shown to be sufficient for u(·, t) - t 0 as t - t 00. We note that in the critical case P = Pc, Theorem 6.1 holds true by replacing the condition (20) by lim Ixlm+>'l (In(lxl))-1 {
uo(x) = max{ <Pa(x), uo(x)}.
319
If Ixl is sufficiently large, then Uo < CPo. by assumption on uo(x) and (3), so that Uo == CPo.. Since Uo :::; uo, by the comparison principle, the corresponding solution satisfies u(·, t; uo) :::; u(-, t, uo) for all t. Since uo, is compactly supported perturbations of the steady states CPa, it follows from Theorem 3.1 that
0:::; limsupu(x,t,uo):::; lim u(x,t,uo) = CPa(x) t-+oo
t-+oo
Since this is valid for arbitrary
0:
and CPo. --; 0 as
0: --;
(x
E
JRN).
0, (20) is proved.
6.2. lJecay rate
Theorem 6.1 does not say anything about the rate of convergence. In the following subsections, we collect results concerning the upper estimate of solutions that decay to zero. Among decaying solutions, a solution of (1) of the form 1
N
1
u(x,t):=(t+1)-p- 1 1J!«t+1)-"2x),
XEJR,
t;:::::O,
(21)
is called a forward self-similar solution. Substituting this in (1), we see that u satisfies (1) if and only if IJ! is a positive solution of
~IJ! + '#. . V'IJ! + _1_1J! + IJ!P = 0, 2
(22)
p-1
It is known (see Subsection 8.1) that this equation has a positive solution if and only if p > pp. Then, by the comparison principle, if Uo satisfies
0:::; uo(x) :::; lJ!(x) the solution u exists globally in time and remains below the self-similar solution. This in particular means that the solution satisfies
IIu(" t)IIv>o(IRN) :::;
for all t > 0
cr1/(p-l)
(23)
with some positive constant C. For more general initial data, Galaktionov and Vazquez 17 showed that the solution of (1) exists globally and satisfies (23). Later, Souplet and Weissler 47 improved this result; (23) holds true for Uo :::; (1 + c:)cp= with sufficiently small c: > o. Recently, PolaCik-Quittner-Souplet 38 proved that if u is defined for all t > 0 and either (X)
u is radial and
pp
< p < Ps
:= {
for N:::;2,
N +2 N - 2 for N>2,
320 or 00 PF
< P < PBV
for N
= 1,
N(N + 2) { (N _ 1)2 for N> 1,
:=
°
then there is C = C(p, N) > such that (23) holds. Thus the convergence to zero is at least of the order of self-similar solutions in the above cases. See also Wang49 and Gui-Ni-Wang 20 for related results. Kawanago 28 considered the subcritical case PF < P < Ps and showed the following: Suppose that g(x) 20,1= 0, is a continuous bounded function on ]RN with compact support, and that the initial value Uo is given by Uo = Ag(X), where A > is a parameter. Then there exists A* E (0,00) such that
°
(i) if A > A*, the solution of (1) blows up in finite time, (ii) if A = A*, the solution of (1) approaches a self-similar solution and hence it decays to zero with the order of C 1/(p-l), and (iii) if < A < A*, the solution of (1) decays to zero with the order of o(C 1/(p-1)).
°
Thus, the decay rate C 1/(p-1) appears for threshold solutions. More precise results were recently obtained by Polacik37 and Quittner43 concerning the behavior of threshold and sub-threshoed solutions.
6.3. Slowly decaying solutions If the initial value decays slowly in space, then there may exist a solution
that decays more slowly than C
Ilu(·, t)lIu"'(lR N)
-+
°
1/(p-l).
A solution of (1) satisfying
1
and tp-l Ilu(" t)llu"'(lRN)
-+ 00
as t
-+ 00
(24)
is called a slowly decaying solution. In this subsection we will deal with the case where P > Pc and Uo satisfies (AI) and
IPoo(x) - c1l x l- 1 :::; uo(x) :::; IPoo(x) - c2lxl- 1
for Ixl > R
(25)
with some constants l > m and C1, C2, R > 0. Recall that if l = m + A!, then the solution remains bounded and bounded away from zero (see Theorem 3.1), and that if l > m + AI, then the solution of (1) becomes unbounded as t -+ 00 (see Theorem 5.1). On the other hand, Gui, Ni and Wang 20 proved that if m < l < m + AI, then the solution of (1) converges to and satisfies (24). Here we give the exact rate of the slow convergence.
°
321
Theorem 6.2 (Fila-Winkler-Yanagida13 ). Let p > Pc and l E (m, m+ AI). Suppose that Uo satisfies (A1) and (25). Then there exist positive constants GI , G2 such that the solution of (1) satisfies
GI(t + 1)for all t
~
m(m+>'l -I) 2,)"
~
lIu(·, t)lloo
~
G2 (t + 1)-
m(m+>'l -I)
2X,
o.
This theorem particularly implies that any rate slower than the selfsimilar one occurs for some initial data. Combining this theorem with the results in Subsection 5.2 on the growup rate of solutions, we see that the estimate in the above theorem holds true for a wider range l E (m, m + A2 + 2). In fact, our method of the proof of Theorem 6.2 can be applied also to l E (m + AI,m + A2 + 2), and it unifies and simplifies the proof for the rate of the convergence to zero and grow-up rate. Our idea to prove Theorem 6.2 is to use the self-similar change of variables , x v(y,s) = (t+1)v-'u(x,t), s = log(t + 1), y = v't + l' and transform (1) into
vs = ~v + ~ . 'Vv + p~l V
+ vP
{ v(y,O) = uo(y).
Y E ]RN,
S
> 0, (26)
Notice that the self-similar solution given by (21) corresponds to a steady state of this equation. Then any solution of (1) converging slowly to zero corresponds to a grow-up solution of the transformed equation. We determine the grow-up rate by using a comparison method. An advantage of using the self-similar variables is that for the construction of suitable super- and subsolutions, we may use functions with separated variables, which makes analysis much easier and more transparent.
7. Convergence to self-similar solutions
7.1. Forward self-similar solutions In this section, we consider the convergence of solutions to radially symmetric forward self-similar solutions
322
where '¢a is a positive solution of '¢pp + N;l'¢p + ~'¢p { '¢(O) = a, '¢p(O)
+ rq.'¢ + J'¢JP-l,¢ =
0,
= 0,
p> 0,
(27)
where a > 0 is a parameter. The self-similar solutions are known to play an important role in the dynamics of solutions of (1). In this subsection, we summarize known results concerning the properties of forward self-similar solutions. It was shown by Haraux-Weissler 24 that the initial value problem (27) has a unique global solution '¢a, and '¢a satisfies
where La is a constant depending on a. If '¢a(P) is positive for all P > 0 then U = ua(x,t) solves (1) with uo(x) = '¢a(JxJ). Concerning the existence and uniqueness of positive solutions of (27), it is found by Haraux-Weissler,24 Yanagida50 and Dohmen-Hirose 5 that the Fujita exponent and the Sobolev exponent are critical.
Theorem 7.1 (Dohmen-Hirose 5 and Yanagida 50 ).
(i) If 1 < p :::; PF, then '¢a vanishes at some finite p for any a> O. (ii) If PF < P < Ps, then there exists ao(p) with the following properties: For a E (O,ao(p)), '¢a(P) is positive for all p > 0 and La > O. For a = ao(p), '¢a(P) is positive for all p > 0 and La = O. For a> ao(p), '¢a(P) vanishes at some finite p. (iii) If p ~ Ps, then for any a > 0, '¢a(P) is positive for all p > 0 and La> O.
We note that for every p > PF, 'l/Ja converges uniformly to the trivial solution '¢ == 0 as a 1 O. Recently, Naito 35 showed that a structural change of the set of selfsimilar solutions occurs at some critical exponents. According to his results, if p > PF then the family of solutions of (27) is ordered at least for small a > O. More precisely, a*(p):= sup {Ci : '¢a(P) is monotone increasing in a E (0, Ci)for every r E [O,oo)}
satisfies a*(p)
E(O,OO) ifPF
323 In particular, if p ~ Pc, the family of solutions of (27) is totally ordered for all a > 0, and as a ---- 00, 'l/Ja(P) converges to a singular solution p> O.
For convenience, we set
'I/J*(p) := {'l/Ja.(p) for p
E
(PF,Pc),
'l/Joo(p) forpEfpc,OO). 7.2. Convergence rote In this subsection we consider solutions of (1) which converge to self-similar solutions as t ---- 00. We introduce the following auxiliary problem:
Vpp
+ N;lVp + ~Vp + ~V + p('l/Ja)P-1V = 0,
{ V(O) = 1,
p > 0, (28)
Vp(O) = 0,
where 1 E lR is a parameter. As we will see later, for each a E (0, a*(p)) there is a unique la (> m) such that the solution of (28) with 1 = la is positive for all p > 0 and satisfies V(p) = o(p-l) as p ---- 00. (See Theorem 7.5 below for a more precise description.) We may regard la as the principal eigenvalue of (28). Now we state a result on the convergence rate.
Theorem 7.2 (Fila-Winkler-Yanagida 14 ). Let p > PF and a (0, a*(p». Suppose that Uo satisfies
E
o :S uo(x) :S ¢*(Ix[), and for Ixl > R with some positive constants l, k 1, k2, R.
(i) If l E (m, La) then there exists a constant G > 0 such that the solution of (1) satisfies
Il
u(" t) - (t
+ 1)- P-=-l 'l/Ja((t + 1)-~ I· 1)11 LOO(JRN) ~ G(t + 1)-~
for all t > O. (ii) If l E [la, 00) then for any e > 0, there exists a constant Gf: > 0 such that the solution of (1) satisfies
Il
u(., t) - (t
for all t > O.
+ 1)- P-=-l 'l/Ja((t + 1)-~ I· [)II Loo (JRN) ~ Gf:(t + 1)-~+f:
324 The next result gives a lower bound which shows that the upper estimate in Theorem 7.2 (i) is optimal.
Theorem 7.3 (Fila-Winkler-Yanagida14 ). Let p > PF, a E (O,a*(p)) and l E (m, La). Suppose that Uo satisfies either
'l/>OI(lxl):::; uo(x) :::; '1/>* (Ixl) for Ixl > 0, { uo(x) :::: 'l/>OI(lxl)
+ b+lxl- l
for Ixl > R,
or
0:::; uo(x) :::; 'l/>OI(1XI) for Ixl > 0, { uo(x):::; 'l/>OI(1XI) -'- Llxl- l for Ixl > R with some positive constants b± and R. Then there exists a constant c > such that the solution of (1) satisfies
Iu(-, t) l
(t
°
+ 1)- p':l 'l/>0I«t + 1)-! 1'1)11 Loo(JRN) :::: c(t + 1)-4
for all t > 0. If we only assume that uo is below or above '1/>01 then we have the following universal estimate, which implies that Theorem 7.2 (ii) is optimal.
Theorem 7.4 (Fila-Winkler-Yanagida 14 ). Let p > PF and a (O,a*(p)). Suppose that uo satisfies uo =1= '1/>01 and either
'l/>OI(lxl) :::; uo(x) :::; '1/>* (Ixl)
for Ixl
>
E
°
or
Then for any c > (1) satisfies
Iu(.,t) l
(t
°
there exists a constant
for Ixl Cc
>
°
> 0.
such that the solution of
+ l)-P':l'1/>OI«t + 1)-!1'1)11 Loo (JRN) :::: cc(t + l)-~-C
for all t > 0. In the above theorems, the principal eigenvalue La plays an important role. We have the following result concerning the dependence of La on a.
Theorem 7.5 (Fila-Winkler-Yanagida14 ). Let p > PF. Then La has the following properties:
(i) La is continuous and strictly decreasing in a
E (O,a*(p)).
325
(ii) la i N as a 1 O. (iii) IfpF < P < Pc, then la 1 m as a i a*(p). (iv) If p ~ Pc, then la 1 m + ).2 + 2 as a i 00. For the proofs of Theorems 7.2, 7.3 and 7.4, we first rewrite (26) in the radial setting vs {
= vpp + ~vp +
N;l
+ P~l V + v P
y E ]RN,
S
> 0, (29)
v(p,O) = uo(p),
where p = Iyl. Using a result of the spectral analysis of the linearized operator around the self-similar solution, we construct suitable sub- and supersolutions of (29). Theorem 7.5 can be proved by an ODE approach.
8. Rapidly decaying solutions
8.1. Linear behavior of solutions A solution of (1) satisfying 1
tp-l
Ilu(·, t)/IV"'(IRN) --.0
as t --.
00.
(30)
is called a rapidly decaying solution. First we give two different sufficient conditions on Uo and p which guarantee that (30) holds.
Theorem 8.1 (Fila-King-Winkler-Yanagida10 ). Suppose that (A 1) holds If for Ixl > R with some positive constants l global and satisfies (30).
> m,
(31)
k and R. Then the solution of (1) is
Another sufficient condition for (30) is given as follows.
Theorem 8.2 (Fila-King-Winkler-Yanagida10 ). Let p > PF, and let
326
with some I ~ m and sufficiently small positive constants Cl, C2 then the solution of (1) exists globally in time and there are constants Gl , G2 > and T > 1 such that
°
where
C~
if 0< I < N,
C:lnt ifl=N,
g(t):=
{
if I> N.
CT
This result says that the solution of (1) decays to zero at the same rate as the solution of the linear heat equation
x
E ]RN,
t > 0,
(34)
with the same initial condition U(x,O) = uo(x). In this subsection we answer the question of whether or not the linear behavior (33) holds for solutions decaying to zero if Uo satisfies (32) for some I > m but the constant C2 is not necessarily small.
Theorem 8.3 (Fila-King-Winkler-Yanagida10 ). Let p > PF. Suppose that Uo satisfies (32) with some positive constants I > m, Cl, C2 (not necessarily small) and R. Suppose further that u is global and satisfies (30). Then there are constants Gl, G 2 > and T > 1 such that (33) holds.
°
The lower bound in (33 ) is trivial because of the positive source term. The upper bound in (33 ) can be obtained by constructing suitable supersolution.
8.2. Convergence to self-similar solutions of the linear heat equation In this subsection, assuming more precise behavior of Uo as Ixl -+ 00, we show that the solution of (1) converges to a self-similar solution of the linear heat equation. Let W = Wl(p) be the unique solution of wpp
N-l + E.2wP + 2I w + -p-wp
{ w(O) = 1,
wp(O)
--
° ,
p> 0,
= 0.
By direct computation, we can show that U
= WI(X,t):= (t + 1)-~WI(t+ 1)-~lxl)
(35)
327 is a solution of (34) with the initial value Wt(x, 0) = wl(lxi). We call such a solution a (forward) self-similar solution of the linear heat equation. The following properties of solutions of (35) are well-known (see, e.g., Abramowitz-Stegun 1 ).
(i) If 0 < l < N, then Wl > 0 for all p > O. Moreover there exists a positive constant C[ such that Wl(p) = Clp- l + o(p-l) as p ~ 00. (ii) If l = N, then Wl is written explicitly as Wl(p) = e- p2 / 4 . (iii) If l > N, then Wl changes its sign. Our aim is to show that for l E (m, N), if Uo is close to kWl with some positive constant k for Ixl "" 00, then the solution u of (1) approaches Wl(X, t) as t ~ 00. Here we introduce the following assumption:
(A2) p > PF, and Uo satisfies 0 ::::; uo(x) ::::; 'l/Ia:(/xi) for all x 0:
> 0 with
0:
E jRN
for some
> o.
Our first result is stated as follows.
Theorem 8.4 (Fila-Winkler-Yanagida15). A ssume that (A J) or (A 2) holds. If
Ixll{uo(x) - kWl(X)} ~ 0
as Ixl ~
00
for some l E (m, N) and k > 0, then (t + l)t IIu(·, t) - kWl(·, t)llu"'(lRN) ~ 0
as t ~
00.
The above theorem applies to the case when uo(x) behaves like kWl(lxi) for large lxi, regardless of the difference between uo(x) and kWl(lxi). If the behavior of Uo up to a second order term is specified, then we can determine the exact convergence rate as follows.
Theorem 8.5 (Fila-Winkler-Yanagida15 ). Assume that (AJ) or (A2) holds. Assume further that Uo satisfies
for some l E (m, N), A > l and positive constants k and
(i)
K,.
If l < A < min{pl- 2, N}, then for any c > 0 there exists C£ > 0 such that lIu(·, t) - kWl(-, t)IILoo(lRN) ::::; C£(t
for all t 2:: O.
+ 1)-4+£
328
(ii)
If m < l ~ (N + 2)jp and A;::: pl - 2, then for any € > 0 there exists C e > 0 such that
lIu(., t) - kWl(-, t)IILoo(lRN) ~ Ce(t + 1)-~+e (iii)
for all t ;::: O. If (N + 2)jp < l < N and A ;::: N, then for any C e > 0 such that
Ilu(·, t) -
kWI (-, t)IILoo(lRN) ~ Ce(t
€
> 0 there exists
+ l)-if+ e
for all t ;::: O. The following lower estimates show that the convergence rate given in Theorem 8.5 is sharp.
Theorem 8.6 (Fila-Winkler-Yanagida15 ). Letp> pp, l E (m,N) and let u be a global solution of (1).
(i) If x for some A E (l, N), k > 0 and that
K,
E]RN
> 0, then there exists c > 0 such
for all t ;::: O.
(ii) If
for some k
> 0, then there exists c > 0 such that
Ilu(" t) -
kWl(-, t)IILoo(lRN) ;::: c(t + 1)-
~ 2
for all t ;::: O.
(iii) If
for some k > 0, then for any
Ilu(" for all t ;::: O.
€
> 0 there exists Ce > 0 such that
t) - kWI(-, t) II Loo (lRN)
;:::
ce(t + l)-if- e
329 Note that the rate is not affected by the nonlinearity for parameters in some range, while in a complementary range, the rate depends explicitly on the nonlinearity. Because of the source term uP, the lower estimate is trivial by the comparison argument, while the upper estimate becomes the most technical part. To overcome a difficulty in the upper estimate, a bootstrap argument is used in Fila-Winkler-Yanagida15 to obtain an optimal estimate. Acknowledgments
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332
SPATIAL DYNAMICS OF SOME EVOLUTION SYSTEMS IN BIOLOGY Xiao-Qiang Zhao*
Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NL AtC 5S7, Canada E-mail: [email protected] We first give a brief review on traveling waves, spreading speeds, and global stability for monotone evolution systems with monostable and bistable nonlinearities. Then we outline our recently developed theory and methods for general monotone semi flows and certain non-monotone systems, and their applications to some biological models.
Keywords and phmses: Spreading speeds, traveling waves, global stability, monotone semiflows, biological systems. AMS Subject Classification: 35B40, 35K57, 35RlO, 37C65, 37L60, 92B05, 92D25.
1. Introduction
The study of traveling waves and spreading speeds for evolution equations with spatial structure has a history which is at least 70 years long. A solution u(t, x) of an evolutionary system is said to be a traveling wave solution if u(t, x) = U(x-ct) for some function U. Usually, U is called the wave profile, and c is called the wave speed. If, in addition, two limits U (±oo) exist, this solution is also called a traveling wavefront. Monostable and bistable nonlinearities frequently appear in spatially homogeneous systems. In the following, we briefly review these two typical cases. Fisher (1937) [38J considered the following equation Ut
= U xx + u(l - u), x
E
lR, t ::::
o.
*Supported in part by the NSERC of Canada and the MITACS of Canada.
(1.1)
333 He showed that the wave solution u(t, x) = U(x - ct) exists if and only if Icl ~ Cmin = 2, and conjectured that Cmin is the asymptotic speed of propagation of the advantageous gene. Kolmogoroff, Petrowsky and Piscounoff (1937) [66] established the same result with u(l - u) replaced by a function f(u) having the same qualitative properties, and proved that the solution u(t,x) with u(O, x) = H(-x) converges to the monotone (decreasing) traveling wave with speed Cmin in profile. Aronson and Weinberger (1975, 1978) [5, 6] studied a class of reactiondiffusion equations, and confirmed Fisher's conjecture. More precisely, they proved the following result.
Theorem A. Let u(t, x) be a nonzero solution of (1.1) with u(O, x) having compact support. Then the following two statements are valid:
(i) (ii)
lim
u(t,x) = 0, Vc> 2;
lim
u(t,x)=1,VCE(0,2).
t-+oo,lxl2:ct t-+oo,lxl~ct
Let u(t, x) satisfy the properties (i) and (ii) above. For any given p E (0,1), let x~(t) and x~(t) be the most right and left points with u(t, x~(t)) = p, respectively. We can easily show that
lim t-+oo
x~(t) = ±2 t
uniformly for p in any compact interval contained in (0,1). Thus, it is natural to call this c* = 2 as the asymptotic speed of spread (in short, spreading speed). Since these fundamental works there have been extensive investigations on traveling waves, spreading speeds, convergence, uniqueness, minimal wave speeds, and stability for various evolution equations. It is impossible to include all the related papers in our references. The below is a partial list for the study of monostable waves and spreading speeds. Autonomous reaction-diffusion equations: [47], [114, 115], [136], [lO7,
108], [35], [18], [65], [14], [138], [39], [139], [97], [94], [lO5], [129], [158], [46], [72], [154], [141], [71], [155], [87], and references therein. Density-dependent reaction-diffusion equations: [110-113], [89-91], and [88]. Discrete-time systems Un+l = Q[u n ]: [142, 143], [77-80], [67], [145], [75], [62], and [52].
334
Integral and integrodifferential equations: [7], [19], [4], [27], [29], [117, 118], [28], [130-132], [101-104]' [133], [95], [20], [81], and [148]. Time-delayed reaction-diffusion equations: [116], [150], [82], [2, 3], [42], [133], [53], [109], [34], [140], [159], [44], [83], [100], [147] and references therein. Lattice equations: [57, 162], [149], [24, 25], [146], [85], [23], [45], [156], [87], and references therein. Periodic and almost periodic evolution equations: [121], [74], and [60,
61]. Heterogeneous environment models: [62-64, 125], [152]' [58], [15-17], [144], [123], [45], and references therein. Fife and Mcleod (1977, 1981) [36, 37] proved the existence, uniqueness and asymptotic stability of monotone traveling waves of the following scalar reaction-diffusion equation with bistable nonlinearity: Ut
= Uxx + u(l -
u)(u - a), x E JR, t ~ 0,
(1.2)
where a E (0,1). Chen (1997) [22] further extended this result to nonlocal evolution equations, and developed a squeezing technique for the global exponential stability. Theorem B. Equation (1.2) admits a unique (up to translation) monotone traveling wave solution cp(x - ct), and there exists a positive constant J.L > 0 such that for every bounded and uniformly continuous initial Junction 'lj;(.) on JR with the property that limsup'lj;(~) ~-->-oo
<
a
<
liminf'lj;(~), ~-->oo
the corresponding solution u( x, t, 'lj;) satisfies lIu(x, t, 'lj;) - cp(x - ct + s",)11 :::; C",e-J-Lt, \:Ix E JR, t ~ 0, for some constant s'" E JR and C'" > O. Ludwig et al. (1979) [76] presented the following spruce budworm population model with bistable nonlinearity:
oN
(N)
fit = Db..N + rBN 1 - KB
BN2 - A2 + N2'
Here r B is the linear birth rate of the budworm, and K B is the carrying capacity, which is related to the density of foliage available on the trees. The
335
°
term }f1:~2 with A, B > represents predation, generally by birds. Both the refuge equilibrium and the outbreak equilibrium are linearly stable. The below is a partial list for the study of bistable waves. Monotone reaction-diffusion systems: [151]' [139], [96], [106], [99], [54],
[153], [134, 135], [59]. Time-delayed reaction-diffusion equations: [116], [127], [84], and references therein. Integro-differential equations: [13], [22], [9], and [10]. Periodic reaction-diffusion equations: [1], [8]. Almost periodic and nonautonomous reaction-diffusion equations: [119-
121,124], [21]. Lattice equations: [160, 161], [26], [92, 93], [11, 12], [122], [86]. There are also numerous investigations on traveling waves for other types of evolution equations in biology. Among these models are two-species competition type reaction-diffusion systems (see, e.g., [40], [49-51], [70], [43], [73]) and predator-prey type reaction-diffusion systems (see, e.g., [41],
[30-32], [128], [55, 56]). The purpose of this paper is to survey the theory and methods of spreading speeds and traveling waves for general monotone semifiows and certain non-monotone systems, and their applications to some evolution systems in biology, which were previously presented in papers with collaborators.
2. Monotone systems A family of mappings {
Let T be a nonnegative real number and C be the set of all bounded and continuous functions from [-T, 0] x 1t to IRk, where 1t = IR or Z. Clearly, any vector in IRk and any element in the space C := C([-T, 0], IRk) can be regarded as a function in C. Let X be the space of all bounded and continuous functions from 1t to IRk equipped with the compact open topology. For u = (u l , •.• ,uk), V = (VI, ••. ,v k) E C, we write u ~ v(u» v) provided ui(O,x) ~ vi(O,x)(ui(O,x) > vi(O,x)),Vi = 1,,,, ,k,O E [-T,O],X E
336
H; and u > v provided u ;::: v but u =f. v. For any two vectors a, b in IRk or two functions a, bE C, we can define a ;::: (>,») b similarly. For any r E C with r »0, we define Cr := {u E C: r ;::: u ;::: O} and Cr := {u E C: r ;::: u;::: o}. Define the reflection operator R by R[u] (0, x) = u(O, -x). Given y E H, define the translation operator Ty by Ty [u] (0, x) = u(O, x - y). For given 00 E [-T, 0], Xo E Hand W C C, we use the following notations: W(·, xo) := {¢{,xo) E
C: ¢
E
W}, W(Oo,') := {¢(Oo,·) EX: ¢ E W}.
We C is said to be T-invariant if TyW = W for all y E H. Given a function ¢ E C and a bounded interval I = [a, b] C H, we define a function ¢I E C([-T, 0] x I, IRk) by ¢I(O, x) = ¢(O, x). Moreover, for any subset V of C, we define VI := {¢I E C([-T,O] x I, IRk) : ¢ E V}. Let j3 E C with j3 » 0 and Q = (Ql,'" , Qk) : Cfj introduce the following assumptions on Q:
-t
Cfj. We first
(AI) Q[R[ulJ = R[Q[u]], Ty[Q[ulJ = Q[Ty[ulJ, Vu E Cfj, Y E H. (A2) Q: Cfj - t Cfj is continuous with respect to the compact open topology. (A3) One of the following two properties holds: (a) There is a number IE [0,1) such that for any A C Cfj and x E H, O:'({Q[u](·,x) : u E A}) :::; 1000({u(·,x) : u E A}), where 0:' is the Kuratowski measure of noncompactness on the Banach space C. (b) The set Q[Cfj] (0, .) is precompact in X, and there is a positive number <;:::; T such that Q[u](O,x) = u(O+<;,x) for -T:::; 0:::; -<;, and the operator
S[u](O,x):= {U(O,X),-T:::; 0 <-<; Q[u](O, x), -<; :::; 0 :::; 0, has the property that S[D](·,O) is precompact in C for any Tinvariant set D C Cfj with D(O,·) precompact in X. (A4) Q: Cfj - t Cfj is monotone (order-preserving) in the sense that Q[u] ;::: Q[v] whenever u ;::: v in Cfj. (A5) Q : Cfj - t Cfj admits exactly two fixed points 0 and j3, and for any positive number to, there is 0:' E Cfj with Iiall < to such that Q[a] » a. It is easy to see that the hypotheses (A3)(a) holds if {Q[u](·,x) : u E Cfj, x E H} is a precompact subset of C.
337
Theorem 2.1. ([75, THEOREM 2.17] AND [74, REMARK 2.1]) Let {Qtlt~O be a semiftow on C{3 with Qt[O] = 0, QtLB] = (3 for all t 2: 0. Suppose that Q = Ql satisfies all hypotheses (A1)-(A5), and Qt satisfies (AI) for any t > 0. Let c* be the asymptotic speed of spread of Ql. Then the following statements are valid:
°
(i) For any c > c*, if v E C{3 with 0:::; v« (3, and v(',x) = for x outside a bounded interval, then lim Qdv](O, x) = uniformly
°
t-too,jxj~ct
for 0 E [-7,0]. (ii) For any c < c* and 0' E C{3 with 0' » 0, there is a positive number r a such that if v E C{3 and v(·, x) » 0' for x on an interval of length 2ra, then lim Qdv] (0, x) = (3(0) uniformly for 0 E [-7,0]. If, in t-too,jxj:Sct
addition, Ql is subhomogeneous, then r a can be chosen to be independent of 0' » 0.
To estimate the spreading speed c*, we can use the following linear operators approach. Let M : C --+ C be a linear operator. Assume that (MI) M is continuous with respect to the compact open topology. (M2) M is a positive operator, that is, M[v] 2: whenever v > 0. (M3) M satisfies (A3) with C{3 replaced by any uniformly bounded subset ofC. (M4) M[R[u]] = R[M[ull, Ty[M[ull = M[Ty [ull, Vu E C, Y E H. (M5) For some ~ E (0, +00]' M can be extended to a linear operator on the linear space C of all function v E e([ -7,0] x H, jRk) having the form
°
v(O, x)
= VI (0, x)eJ1.1X + V2(0, x)eJl.2X, VI, V2
E
C, J..lI, J..l2 E
(-~, ~),
such that if vn , vEe and Vn (0, x) --+ v( 0, x) uniformly on any bounded set, then M[vn](O, x) --+ M[v](O, x) uniformly on any bounded set. (M6) For any J..l E [0, ~), the linear operator BJ1. : C --+ C, defined by
BJ1.[a](O) = M[ae-J1.X](O,O), VO E [-7,0]' is positive, and there is no such that B~o is a compact and strongly positive linear operator on C. (M7) The principal eigenvalue )"(J..l) of BJ1. satisfies that ),,(0) > l.
Theorem 2.2. ([75, THEOREM 3.10]) Let Q be an operator on C{3 satisfying (AI)-(A5), and c* be its spreading speed. Assume that there is a linear
338
operator M satisfying (M1)-(M7) such that (J.l) := tIn >'(J.l) assumes its minimum value at some J.l* E (0, .0.). Then the following statements are valid:
(1) If Q[u] :::; M[u] for all u E C{3, then c* :::; inf/LE(o,~) (J.l). (2) If there is some TJ E C with TJ » such that Q[u] ::::: M[u] for any u E Cry, then c* ::::: inf/LE(o,~) (J.l).
°
Note that .0. = +00 is assumed in [75, Theorem 3.10]. In the case where .0. E (0, +00), the proof of [75, Theorem 3.10] implies that [75, Theorem 3.10] with inf/L>o (J.l) replaced by inf/LE(o,~) (J.l) is still valid, provided that (M5) holds for all J.lI,J.l2 E (-.0.,.0.) and (J.l) assumes its minimum value at J.l* E (0, .0.). We say that W(B, x - ct) is a traveling wave of {Qt1t~O if W : [-7,0] x IR --. IRk and Qt[W](B, x) = W(B, x - ct), and that W(B, x - ct) connects fJ to if W(', -00) = fJ and We +00) = 0. In order to obtain the existence of the traveling wave with the wave speed c ::::: c*, we need to strengthen the hypothesis (A3) into the following one.
°
(A6) One of the following two conditions holds: (a) For any number r > 0, there exists l = l(r) E [0,1) such that for any D c C{3 and any interval I = [a, b] of the length r, we have o:((Q[D])r):::; lo:(D1 ), where 0: is the Kuratowski measure of noncompactness on the Banach space C([-7, 0] x I, IRk). (b) The set Q[C{3] (0, .) is precompact in X, and there is a positive number <;":::; 7 such that Q[u](B,x) = u(B+<;", x) for -7:::; B:::; -<;", and the operator
S[u](B, x) :=
{U(O, x), -7:::; B < -<;" Q[u](B, x),
-<;" :::;
B :::; 0,
has the property that S[D] is precompact in C{3 for any Tinvariant set D c C{3 with D(O,') precompact in X. It is easy to see that the hypotheses (A6)(a) holds if Q[C{3] is precompact in C{3 with respect to the compact open topology. Moreover, if 1i is discrete, then the hypothesis (A3) on Q implies the hypothesis (A6). Theorem 2.3. ([75, THEOREMS 4.1 AND 4.2] AND [74, REMARK 2.3]) Assume that for any t > 0, Qt satisfies hypotheses (A1)-(A5) and let c* be the asymptotic speed of spread of QI. Then the following two statements are valid:
339
°
(i) For any < c < c*, {Qtlt~O has no traveling wave wee, x - ct) connecting (3 to 0. (ii) If, in addition, Qt satisfies (A6) for any t > 0, then for any c ~ c*, {Qtlt~O has a traveling wave W(e,x - ct) connecting (3 to such that W (e, s) is continuous and nonincreasing in S E R
°
We should point out that Theorems 2.1,2.2 and 2.3 were highly motivated by the earlier works of Weinberger [143], Lui [79] and Li, Weinberger and Lewis [72]. More precisely, the existence of spreading speeds and traveling waves for a scalar discrete-time recursion model on a habitat which may be either continuous or discrete was established in [143]. The time map approach to continuous-time models has been prescribed explicitly in the recent paper [72] for cooperative reaction-diffusion systems. Further, the spreading speed results for a system of discrete-time recursions were given in [79] in the case where the linear operators have compact supports. Note that for a time-delayed reaction-diffusion equation or lattice system, one can show that its solution map Qt satisfies A(6)(a) for t > r, and A(6)(b) with <; = t for t E (0, r], under appropriate assumptions. Remark 2.1. Theorems 2.1 and 2.3 are still valid provided that the interval [-r, 0] is replaced with a compact metric space and that the hypotheses (A3) and (A6) are replaced with (A3)(a) and (A6)(a), respectively. The theory of spreading speeds and traveling waves has been further developed to monotone periodic semiftows in [74]. It should be interesting to extend this theory to almost periodic and general nonautonomous systems. Regarding the global asymptotic stability with phase shift of traveling wave fronts of minimal speed, in short minimal fronts, there is no general result. For the scalar reaction-diffusion equation Ut
= U xx + feu),
(x,t) E lR x (0,00),
(2.1)
this problem was addressed in [87] via the method of upper and lower solutions and a squeezing technique under the following assumptions: (Fl) f E C 1 ([0,I],lR), f(O) = f(l) = 0, 1'(1) < 0, and feu) > O,Vu E (0,1). (F2) There exist two constants L > and II > such that 1f'(Ul) 1'(u2)1 ::; Llul - u2l v , V(Ul,U2) E [0,1]2.
°
°
Theorem 2.4. ([87, THEOREM A']) Assume that (Fl) and (F2) hold. Let (U, c) be a traveling wave of (2.1) such that
340
(Pc) lim x ...... -
oo
reg?
= A2 >
AI, where Al and A2 are two roots of c),
=
+ 1'(0). Then U(x + ct) is globally ),2
exponentially stable with phase shift in the sense that for any a E (Ad A 2 , 1), there exists a constant f > 0 such that for any initial data
x~-oo
the solution u( x, t,
2.2. Bistable case For cooperative reaction-diffusion systems with positive diffusion coefficients, and scalar nonlocal evolution equations, the existence, uniqueness and global asymptotic stability of bistable waves are well-known. However, there is no general result on the existence of bistable waves for monotone semiflows. The methods include the phase space analysis, shooting method, perturbation method, etc. In general, it is more difficult to obtain the existence of bistable waves than monostable waves. The squeezing technique can be used effectively to prove the global asymptotic stability of bistable waves for scalar evolution equations (and their lattice versions). Recently, Tsai [135] also applied this technique to a class of monotone reaction-diffusion systems. A dynamical systems approach was developed in [153, 157] to prove the global attractivity (and hence uniqueness) of bistable waves for monotone systems. For simplicity, we let
Ut(x, t) = duxx(x, t) By the moving coordinate z
+ f(u(x, t».
(2.2)
= x - ct, we transform (2.2) into
Ut(z, t) = cUz(z, t)
+ duzz(z, t) + f(u(z, t».
(2.3)
341
Then
Qt(¢)
=
u(·,t,¢), t 2: 0, ¢
E
BUC(JR,JR),
but {Qth:::o is not strongly monotone. For any interval [a, b] c JR, the ordered arc L:= {
+ s)
: s E [a,b]}
consists of equilibria of (2.3). It then suffices to study the convergence of an orbit of (2.3) to some equilibrium in L. For this purpose, one can use the following convergence result for monotone semifiows. Theorem 2.5. ([157, THEOREM 2.2.4]) Let U be a closed convex subset of an ordered Banach space X, and q,(t) : U - t U be a monotone semifiow. Assume that there exists a monotone homeomorphism h from [0, 1J onto a subset of U such that
(1) For each s E [0, IJ, h(s) is a stable equilibrium for q,(t) : U - t U; (2) Each orbit of q,(t) in [h(O), h(I)]x is precompact; (3) If h(so) <x w(¢) for some So E [0,1) and ¢ E [h(O), h(I)]x, then there exists Sl E (so, 1) such that h(Sl) ~x w(¢). Then for any precompact orbit )'+(¢o) of q,(t) in U with w(¢o) n [h(O), h(I)]x =I 0, there exists s* E [0,1] such that w(¢o) = h(s*).
This approach was used to prove the global attractivity of traveling waves for scalar periodic reaction-diffusion equations in [157, Theorem 10.2.1] and for a class of reaction-diffusion systems in [153, Theorem 3.1]. To obtain the exponential stability of bistable waves for monotone systems, one needs to do spectral analysis (see, e.g., [153, Theorem 4.1]). In the rest of this subsection, we present the recent results obtained in [59J. Let D > 0. Consider a reaction-diffusion system
au at =
EPu
DlJX'!
+ F(u, v),
av at = G(u,v). Assume that
(2.4)
342
(HI) There exist three points E_ = (0,0), Eo = (al' bt) and E+ = (a2' b2) with < al < a2 and < bl < b2 such that
°
°
°
°
(1) F, G E Cl(lR~,lR), Fv(u, v) 2'0, Gu(u, v) 2 and Gv(u, v) < on lR~, and Gu(O, 0) > 0. (2) E_, Eo and E+ are only zeros of feu, v) := (F(u, v), G(u, v» in the order interval [E_, E+]. (3) All eigenvalues of the Jacobian matrices Df(E_) and Df(E+) have negative real parts, and D f(Eo) has an eigenvalue with positive real part and another with negative real part. (4) Fv(u,v) > for (u,v) E [0,a2] x [0,b 2].
°
By the assumption (HI), it follows that the spatially homogeneous system du dt dv dt
= F(u,v), (2.5)
= G(u,v)
has only three equilibria E_, Eo and E+ in [E_, E+], E_ and E+ are stable, Eo is a saddle. By a shooting method, we obtain the following result on the existence of bistable waves of (2.4) connecting E_ and E+. Theorem 2.6. ([59, THEOREM 2.1]) Let (HI) hold. Then system (2.4) has a monotone increasing traveling wave solution (U(x + ct), Vex + ct» connecting E_ to E+ for some real number c such that the wave speed c a2 has the same sign as the integral fo F(U, V*(U»dU, where V*(U) satisfies G(U, V*(U» = 0.
To obtain the global attractivity of bistable waves, we need the following additional conditions on F and G. (H2) F and G can be extended to the domain (-1,00)2 for some I > 0 such that
(1) F,G E C 2((-1,00)2,lR), Fu(u,v) < 0, Fv(u,v) > 0, Gu(u,v) 20 and Gv(u, v) < for (u, v) E (-1,00)2. (2) There exists L > 0 such that for any 12 > L, there exists h > 0 such that F(ll' 12) < O.
°
Let X = BUC(lR, lR 2 ) be the Banach space of all bounded and uniformly continuous functions from lR to lR 2 with the usual supreme norm.
343 Let X+ = {('Ij!l,'1h) EX: 'lj!i(X) 2: O,'v'x E ~,i = 1,2}. By the aforementioned dynamical systems approach as applied to system (2.4), we have the following result.
Theorem 2.7. ([59, THEOREM 3.1]) Assume that (HI) and (H2) hold. Let ¢(x - ct) be a monotone traveling wave solution of system (2.4) and iI!(t,x,'Ij!):= (u(t,x,'Ij!),v(t,x,'Ij!)) be the solution of (2.4) with iI!(0,·,'Ij!) = 'Ij! E X+. Then for any 'Ij! E X+ with limsup'lj!(~)« ~---.-oo
Eo«
liminf'lj!(~),
(2.6)
~---.oo
there exists s..p E ~ such that lim 11iI!(t,x,'Ij!) - ¢(x - ct t---.+oo
+ s..p)IIIR2
=
°
uniformly for x E R Moreover, any traveling wave solution of system (2.4) connection E_ and E+ is a translate of ¢. We remark that by the spectrum analysis as in [153, Section 4], one can obtain the local exponential stability with phase shift of the bistable wave ¢(x - ct) with c -10. This, together with Theorem 2.7, implies the global exponential stability with phase shift of the bistable wave ¢(x - ct) with c -I of (2.4).
°
3. A class of non-monotone systems Many discrete- and continuous-time population models with spatial structure are not monotone. For example, scalar discrete-time integrodifference equations with non-monotone growth functions (see, e.g., [67]), and predator-prey type reaction-diffusion systems are among such models. The spreading speeds were obtained for some non-monotone continuous-time integral equations and time-delayed reaction-diffusion models in [131, 133], and a general result on the nonexistence of traveling waves was also given in [133, Theorem 3.5]. The existence of monos table traveling waves were established for several classes of non-monotone time-delayed reaction-diffusion equations in [34, 83, 100, 150]. As an illustration, below we present the main results obtained recently in [52]. Let C be the space of all bounded and continuous functions from ~ to ~ equipped with the compact open topology. For a given number r > 0, let Cr := {¢ E C: 0::; ¢(x) ::; r, 'v'x E ~}. Let k(x) be a nonnegative Lebesgue measurable function on R We assume that the kernel k(x) has the following property:
(K) fIR k(y)dy = 1, k( -y) = key), 'v'y
E ~,
and
fIR e-nYk(y)dy <
00,
'v'a
E
344
[0, ~), where infinity.
~
> 0 is the abscissa of convergence and it may be
Consider a discrete-time integrodifference equation Un+l(x)
=
1
h(un(y))k(x - y)dy, x E JR., n
~0
(3.1)
with Uo E C. Assume that there exists (3 > 0 such that (B1) h E C([O, (3], [0, (3]), h(O) = 0, h' (0) > 1, h({3) = (3, and there is Lo > 0 such that Ih(ur) - h(u2)1:::; Lolul - u21, 'iUI, U2 E [0,{3]. (B2) U < h(u) :::; h'(O)u, 'iu E (0,{3), and h(u) is nondecreasing in u E [0,{3].
Let U(x) be a continuous function on lR. We say U(x+en) is a traveling wave solution of (3.1) with the wave speed c if un(x) = U(x + en), 'in ~ 0, satisfies (3.1), and U(x+cn) connects 0 to (3 if U( -00) = 0 and U( +00) = {3. Define
(3.2) The following result is essentially due to Weinberger [143], and shows that is not only the spreading speed but also the minimal wave speed of monotone traveling waves for system (3.1).
c;,
Theorem 3.1. ([52, THEOREM 2.1]) Let (B1) and (B2) hold. Then the following statements are valid: (i) For any Uo E C{3 with compact support, the solution of (3.1) satisfies lim un(x) = 0, 'ic > c;'. n-+oo,lxl~cn
(ii) Foranyuo EC{3\{0},thesolutionof(3.1) satisfies
lim
n-+oo,lxl~cn
un(x)
=
{3, 'ic E (0, c;,). (iii) For any c ~ c;', (3.1) has a traveling wave U(x + cn) connecting 0 to (3 such that U(x) is nondecreasing in x, and for any c E (0, c;'), (3.1) has no traveling wave U(x + cn) connecting 0 to (3. Now we consider the discrete-time integrodifference equation Un+l(X)
=
1
f(un(y))k(x - y)dy,
x E JR., n
with Uo E C. Assume that there exists b > 0 such that
~0
(3.3)
345
(Dl) f E G([O, b], [0, b]), f(O) = 0, f'(O) > 1, and there is L > 0 such that If(ut} - f(U2)1 ~ Llul - u21, VUl, U2 E [0, b]. (D2) f(u) ~ f'(O)u, Vu E [0, b], and there is u* E (0, b] such that f(u*) = u*, f(u) > u, Vu E (O,u*), and 0 < f(u) < u, Vu E (u*,b]. Three types of growth functions are commonly used in population biology: logistic type function f(u) = ru (1- *"), r > 0, K > OJ the Ricker type function f(u) = que- PU , q > 1, p > OJ and the generalized Bevertonm > 0, and p > q > o. Holt type function f(u) = Define
q!:m,
f+(u)
=
max f(v),
O~V~U
f-(u)
=
min f(v),
u~v~b
Vu E [0, b].
It then follows that
that both f+ and f- are nondecreasing and Lipschitz continuous, with the Lipschitz constant L, on [0, b], and that there exists 00 E (0, b] such that f±(u) = f(u), Vu E [0,00]. Let u± be such that f±(u±) = u±. Then o < u~ ~ u* ~ u+ ~ b. To obtain the upward convergence as stated in Theorem 3.1(ii), we need to impose one of the following two additional conditions on f. (Cl) u* = band f(u) is nondecreasing in u E [b- EO, b] for some EO E (0, b). (C2) fC;:) is strictly decreasing for u E (O,b], and f(u) has the property (P) that for any v, w E (0, b] satisfying v ~ u* ~ w, v ~ f(w) and w ~ f(v), we have v = w. It follows from [52, Lemma 2.1] that either of the following two conditions is sufficient for the property (P) in condition (C2) to hold: (PI) uf(u) is strictly increasing for u E (0, b]. (P2) f(u) is nonincreasing for u E [u*,b], and f2~u) is strictly decreasing for u E (0, u*]. The following two results were proved in [52] via the comparison methods, the Schauder fixed point theorem, and the limiting arguments. Theorem 3.2. ([52, THEOREM 2.2]) Let (Dl) and (D2) hold and cj be defined as in (3.2) with h = f. Then the following statements are valid:
+
(i) For any Uo E Cu with compact support, the solution of (3.3) satisfies lim un(x) = 0, Vc> cj. n-+oo,lxl~cn
346
(ii) For any Uo E Cu +\ {O}, the solution of (3.3) satisfies u:':S
liminf
n->oo,lxl~cn
un(x):S
limsup
n->oo,lxl~cn
un(x):S u~, 'tic E (O,ej).
(iii) If, in addition, either (C1) or (C2) holds, then for any Uo E Cu +\ {O}, the solution of (3.3) satisfies
lim
n->oo,lxl~cn
un(x)
= u*, 'tic E (O,ej).
We should point out that Theorem 3.2 (iii) in the case of (C2) and its proof were highly motivated by [131, Lemma 3.10) and [133, Theorem 2.5) on continuous-time integral equations. Remark 3.1. Theorem 3.2 with fIR replaced by fIRm is still valid under the assumption that fIRm k(y)dy = 1, k(x) = key), 'tIx, y E ]Rm with Ixi = Iyl.
Note that if 1"(0) exists, then feu) 2:: f'(O)u - au 2 , 'tIu E [0,0), for appropriate a > 0 and 0 > O. To obtain the existence of traveling waves, we impose the following weaker condition on f. (D3) There exist real numbers 0* E (0,00), a feu) 2:: f'(O)u - au ff , 'tIu E [0,0*].
Theorem 3.3. ([52, THEOREMS 3.1 the following statements are valid:
AND
> 1 and a > 0 such that
3.2)) Let (D1)-(D3) hold. Then
(i) For any e E (0, ej), (3.3) has no traveling wave U(x + en) with U E Cu {O} and U( -00) = O. (ii) For any e > ej, (3.3) has a traveling wave U(x + en) such that U E Cu {O}, U(-oo) = 0 and
+\ +\
u:':S liminfU(~):S limsupU(~):S u~. e->+oo e->+oo If, in addition, either (C1) or (C2) holds, then U( +00) (iii) (3.3) has a traveling wave U(x + ejn) such that U E Cu
u:. :S
= u*.
+\ {O, u*} and
liminfU(~):S limsupU(~) :S u~.
e->+oo
e->+oo
If, in addition, either (C1) or (C2) holds, then U(+oo) = u*.
In view of Theorems 3.2 and 3.3, we see that the spreading speed is linearly determinate and coincides with the minimal wave speed of traveling waves for this class of non-monotone discrete-time integrodifference equation population models.
347 4. Applications to biological systems The invasion speed is a fundamental characteristic of biological invasions, since it describes the speed at which the geographic range of the population expands, see, e.g., [48, 68, 69, 98, 126, 137] and references therein. In this section, we choose five biological models to illustrate the applicability of the theory and methods mentioned in the previous sections.
4.1. A model with a quiescent stage Hadeler and Lewis (2002) [46] presented and discussed briefly the following model
+ f(ur)
Ot U1 = D.6.U1
- '/'l U1 + '/'2U2,
Ot U2 = '/'1 U1 - '/'2 U2,
(4.1)
which describes a population where the individuals move between mobile and nonmobile states, and only the migrants reproduce. Such behavior is typical for invertebrates living in small ponds in arid climates which dry up and reappear subject to rainfall. Assume that
> 0, e~V»)' < :::; 0 for all v ? H.
(E1) f E C 1 (JR.+,JR.), f(O) = 0,1'(0) exists H > 0 such that f(v)
0 for v
> 0, and there
Then (4.1) has a unique positive constant solution u * . Define >'(JL) =
~
[DJL
2
+ 1'(0) -
1
'/'1 - '/'2]
+ 2V(DJL 2 + 1'(0) -
'/'1 - '/'2)2
+ 4'/'2(DJL2 + 1'(0)).
Note that the solution maps associated with (4.1) are not compact with respect to the compact open topology, but they satisfy assumption (A6)(a). By the theory of spreading speeds and monostable traveling waves for monotone systems, we then have the following result. Theorem 4.1. ([155, THEOREMS 2.1 AND 2.2» Assume that (E1) holds, and let c* = inf A(JL). Let u(t, x, 4» be the solution of (4.1) with u(O", 4» = JL>O
4>
E Xu'
JL
= C(JR., [0, u*)).
Then the following statements are valid:
(i) For any c > c*, if 4> E Xu' with 0 :::; 4> « u*, and 4>(x) outside a bounded interval, then lim u(t,x,4» = 0; t->oo,lxl~ct
=
0 for x
348
(ii) For any c E (0, c*), if ¢ E Xu' and ¢ ¢ 0, then
lim
t.-oo,lxl::Sct
u(t, x, ¢) =
u*. (iii) c* is the minimal wave speed for monotone traveling waves of (4.1) connecting and u*.
°
By Theorems 2.6 and 2.7, it is easy to see the following result on the bistable wave is valid. Theorem 4.2. ([59,
EXAMPLE
3]) Assume that
°
°
(E2) There exists l > such that f E C 2( -l, 00) and f'(ud -1'1 < for U1 E (-l,oo), and f(ur) has only three zeros 0< a1 < a2 on the interval [0,a2] with 1'(0) < 0, I'(ad > and f'(a2) < 0.
°
Then system (4.1) admits a bistable traveling wave, which is globally attractive with phase shift and unique up to translation.
4.2. A nonlocallattice differential system Weng, Huang and Wu (2003) [146] derived a mature population growth model
dWj(t)
~
= D[wj+1 (t) + Wj-1 (t) - 2wj(t)]- dWj(t)
+~
00
L
(3Q(j - k)b(wk(t - r»,
(4.2)
k=-oo
where t > 0, j E Z,
(3Q(l) = 2e- v r
~
17r cos(lw)eVCOSWdw,
0, D, d, /-L and v = 2a are all positive real numbers. Assume that
(E3) bE C(lR+,IR+), b(O) = 0, b'(O) > d//-L, b(w) S b'(O)w for W E IR+. (E4) b(·) is strictly increasing on [0, K] for some K > 0, and /-Lb( w) = dw has a unique solution w+ E (0, K]. The authors of [146] also proved the existence of spreading speed c* and the existence of traveling waves with wave speed c > c* in the case where the time delay r is small. The following result gives a complete description of spatial dynamics of (4.2). Define
f(c,x) := cx - [D(e- X + eX) - (d + 2D)]_/-Lb'(0)e(cosh x -1)V-CXT.
349
Let (c*, x*) be the positive solution to the following system
of
ox (c,X)
f(c,x) =0,
=
o.
Theorem 4.3. ([75, THEOREMS 5.3 AND 5.4]) Assume that (E3) and (E4) hold. Let wet) be a solution of (4.2) with 0 ~ Wi(t) < w+ for any t E [-r, 0], i E Z. Then the following statements are valid;
(i) If Wi(t) lim
=
t-+oo,lil2:ct
0 for t E [-r, 0] and i outside a bounded interval, then Wi(t) = 0 for any c > c*.
(ii) If wet) ¢ 0 for t E [-r,O], then
lim
t-+oo,lil~ct
Wi(t)
= w+
for any c
< c*.
(iii) For any c ~ c*, (4.2) has a traveling wave solution Wi(t) = U(i ct) such that U(s) is continuous and nonincreasing in s E JR, and U( -00) = w+ and U( +00) = O. Moreover, for any c < c*, (4.2) has no traveling wave U(i - ct) connecting w+ to O.
4.3. A multi-type SIS epidemic model Rass and Radcliffe (2003) [104] presented the following spatial epidemic model
(4.3) Here Yi(X, t) is the proportion of individuals for the ith population at position x who were infectious at time t, lLi ~ 0 is the combined death, emigration and recovery rate for infectious individuals, (Ji ~ 0 is the population size of the ith population, Aij ~ 0 is the infection rate of a type i susceptible by a type j infectious individual, and Pij(U) is the corresponding contact distribution. Let A ;= «(JjAij)nxn, and r = (diag(IL»-lA in the case where IL = (ILl, ... ,lLn) » O. Define
per) ;= max{IAI ; det (AI - r) = OJ. Assume that (E5) Either lLi = 0 for some i, or IL
»
0 and
per) > 1.
350 It was proved in [104] that the spatially homogeneous system associated with (4.3) dYi(t)
~
=
~
(1 - Yi(t)) ~ ajAijYj(t) - /-liYi(t),
1 ~ i ~ n,
j=l
admits a unique equilibrium y* » 0, which is globally asymptotically stable in [0, l]n \ {O}. The open problem on the asymptotic speed of propagation of infection and traveling waves for model (4.3) was solved by Weng and Zhao (2006) [148]. Define a matrix A(a) = (Aij(a))nxn, where ajAij Aij(a)
=
{ ajAij
(1 eOUPij(U)dU) - /-li, (
1
i = j, i =J j.
eOUPij(U)dU) ,
Theorem 4.4. ([148, THEOREMS 3.1-3.2 AND 4.1-4.2]) Assume that (E5) holds. Let A(a) be the principal eigenvalue of A(a), and define c* := inf ),(0). Then c* is the spreading speed for solutions of (4.3) with initial 0>0
0
functions having compact supports. Moreover, c* is also the minimal wave speed for monotone traveling waves of (4.3).
By using the general theory of spreading speeds and traveling waves, Zhang and Zhao (2008) [156] studied the spatially discrete version of (4.3):
aYma~,t)
= (l-Ym(x,t)) tanA mn
1
Yn(x-u,t)Pmn(u)dU-/-lmYm(x,t),
IR
n=l 00
where j E Z, 1 ~ m ~ r,
2:
Pmn(k)
k=-oo
0, Yk E Z, 1 ~ m, n
=
1, and Pmn(k)
=
Pmn( -k) 2::
~ r.
4.4. A vector disease model with spatial spread Ruan and Xiao (2004) [109] presented a diffusive and time-delayed integrodifferential equation au
at (t, x)
= d.6.u(t, x) - au(t, x)
·ft-ooJor
+ b[l -
u(t, x)]
F(t,s,x,y)u(s,y)dyds.
(4.4)
351
Here u(t, x) is normalized spatial density of infectious host at time t and at point x, x is in a spatial habitatn C lRn(n ~ 3), d is the diffusion constant, ~ is the Laplacian operator, a is the cure/recovery rate of the infected host, b is the host-vector contact rate, and F(t, s, x, y) is the convolution kernel, which is positive, continuous in its variables t E lR, s E lR+ and Borel measurable in its variables x, yEn. In the case where n = lR and F(t, s, x, y) = o(x - y)G(t - s) with o(x) being the Dirac o-function and G(t) = f,e- t / r , it was showed in [109] that for any Co 2: 2y'b - a, there exists a small number ro = roCco) > 0 such that for any r E [0, roJ, the model system admits a traveling wave connecting two equilibria 0 and 1 - alb with the wave speed c = c(r) close to Co. By the theory of spreading speeds and traveling waves, the finite delay approximations method, and the limiting arguments, Zhao and Xiao (2006) [159] established the existence of the spreading speed of the disease and the minimal wave speed of monotone traveling waves for the model (4.4) with F(t, s, x, y) = F(t - s, x - y), that is,
au at (t, x) =
d~u(t,
x) - au(t, x)
.[too [: F(t -
+ b[l -
u(t, x)]
s, x - y)u(s, y)dyds.
(4.5)
We assume that
oo
(E6) b> a > 0, F(s, x) = F(s, -x), and fo f~oo F(s, y)dyds = 1. (E7) fooo f~oo F(s,y)e>,(y-cs)dyds < 00 for all c 2: 0 and A 2: 0. Let T > 0 be a parameter. Consider the following reaction-diffusion equation with finite time delay T:
au
at (t,x) = d~u(t,x) - au(t,x) + b[l- u(t,x)] . rJoo F(s,x-y)u(t-s,y)dyds.
Jo
(4.6)
-00
By the theory in Section 3, we can shown that system (4.6) admits a spreading speed c;, which is also the minimal wave speed for monotone traveling waves of (4.6). For c 2: 0 and A 2: 0, define
P(c, A)
=
dA2 - CA - a + b
roo Joo F(s, y)e>,(y-cs)dyds.
Jo
-00
352 It follows that there exists a unique positive solution (c*, A*) to the system
P(C,A)
= 0,
aP
aA (c, A)
= 0.
By the comparison method and the results for integral equations in [133], we can further prove that lim c; = c* . r-->oo
Theorem 4.5. ([159, THEOREMS 2.1 AND 3.1]) Assume that (E6) and (E7) hold and let c* be defined as above. Then c* is the spreading speed for solutions of (4.5) with initial functions having compact supports. Moreover, c* is also the minimal wave speed for monotone traveling waves of (4.5). The finite delay approximations approach was also used in [33] to study the spreading speed and traveling waves for a nonlocal and time-delayed reaction-diffusion population model with age structure.
4.5. A nonlocal and periodic model with dispersal Consider a periodic integro-differential equation
au(t, x) at
[
= F(t, u(t, x)) + a(t) JR k(x - y)u(t, y)dy,
(4.7)
where u( t, x) is the spatial density of a population at the point x E JR at time t ?: 0, F(t, u(t, x)) is the reaction function which governs the population dynamics such as birth and death, and other removal terms such as emigration of individuals at the point x E JR at time t ?: 0, a(t) ?: is the rate at which an individual leaves its current location at time t ?: 0, k(x, y) is the dispersal kernel that describes the probability that an individual moves from point y to point x. Moreover, two continuous functions F and a are w-periodic in t for some w > 0, and a(t) t= 0. For simplicity, we neglect the birth and death of the population during the dispersal process and assume that k(x, y) depends only on the distance between x and y, and then write it as k(x - y). Assume that
°
(E8) F(t, u) = ug(t, u) with 9 E C(JR~, JR) and gu(t, u) < 0, V(t, u) E JR~, foW(g(t, O) + a(t))dt > 0, and there exist u > and L > such that g(t, u) + a(t) ::; 0, Vt ?: 0, and IF(t, Ul) - F(t, u2)1 ::; Llul - u21, Vt ?: 0,Ul,U2 E W:= [O,U]. (E9) k(y) ?: 0, k( -y) = k(y), k(y)dy = 1, and the integral k(y)eQYdy converges for all 0: E [0, ~), where ~ > is the abscissa of convergence and it may be infinity.
°
fR
°
°
fR
353 It is easy to show that the spatially homogeneous system
duet)
d,t = F(t, u(t)) + a(t)u(t)
(4.8)
has a positive w-periodic solution u*(t), which is globally asymptotically stable in [0, u] \ {O}. Define
A(a, t) := get, 0) + aCt)
1
k(y)eQYdy
and W
"') . = Jo A(a, s)ds, <1>( '-'
a
Theorem 4.6. ([60, inf
THEOREM
Va E
(O,~).
3.1]) Assume that (E8) and (E9) hold and
4>(0:)
let c* = o<"'<~ . Let u(t,x,cp) be the solution of (4.7) with u(O,·,cp) cp E Cu*(O) := C(IR, [0, u*(O)]). Then the following statements are valid:
=
(i) For any c > c*, if cp E Cu*(O) with cp(x) = 0 for x outside a bounded interval, then lim u(t, x, cp) = O. t-->oo,lxl:2:ct
(ii) For any 0 < c < c*, there is a positive number r such that if cp E Cu*(O) with cp(x) > 0 for x on an interval of length 2r, then lim (u(t, x, cp) - u*(t)) = o. t-->oo,lxl ~ct
(iii) In the case where aCt) > 0, Vt E IR, for any c E (0, c*), if cp E with cp ¢ 0, then lim (u(t, x; cp) - u*(t)) = O.
Cu*(O)
t-->oo,lxl~ct
Recall that u(t, x) = U(t, x + ct) is an w-periodic traveling wave of (4.7) connecting 0 to u*(t) if it is a solution of (4.7), U(t,~) is w-periodic in t, and U(t, -(0) = 0 and U(t, (0) = u*(t) uniformly for t E [O,w].
Theorem 4.7. ([60, THEOREM 4.1]) Assume that (E8) and (E9) hold. Let c* be as defined in Theorem 4.6. Then for any c E (0, c*), system (4.7) admits no w-periodic traveling wave solution 4>(t, x + ct) connecting 0 and u*(t).
It is reasonable to expect that (4.7) has periodic traveling waves with the wave speed c ~ c* , that is, the above c* is also the minimal wave speed for the monotone periodic traveling waves. Note that we can not use the afore-mentioned general theory to obtain the existence of periodic traveling waves since the solution maps associated with (4.7) are not compact with respect to the compact open topology. However, we have an affirmative
354
answer to this problem in the autonomous case of (4.7) by the method of upper and lower solutions.
°
°
(E8)' F(O) = 0, F"(O) exists, F'(O) + a> and there is u* > such that u* is the unique positive zero of the function F(u) + au in [0, u*], F is Lipschitz continuous on W := [0, u*] with the Lipschitz constant L > 0, and that F(u) :::; F'(O)u for all u E [0, u*]. Theorem 4.8. ([60, THEOREM 4.2]) Let F(t, u) = F(u), aCt) = a, and assume that (E8)' and (E9) Hold. Let c* be defined in Theorem 4.6. Then for any c ~ c*, system (4.7) has a traveling wave ¢(x + ct) connecting to u* such that ¢( s) is continuous and nondecreasing in s E IR..
°
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