oo, fe
+ Cm\\f\\p Cm(a, b), 0 < e < b — a and 0 < j
<m.
PROOF Let Am denote the assertion of the Lemma. AQ and A\ are obviously true; A 3 and that A m _i is true. Note that Am-\ implies ( £ / 2 ) m - 1 | | / ( m - 1 ) | | P < ( e / 2 r | | / ( m ) | | p + C m _ 1 (e/2)||/'|| P e ^ l l / ' l l p ^ C r f r - ' l l / ^ - ^ l l p + Cm-ill/llp for 8 G (0,1). Hence, choosing 8 so that (28)m~2Cm_i
< 1/2 implies
for some constant C. This, together with A m _i, imply Am.
□
Definition 3.4.13 For integers m > 0, p G [l,oo] and those u G L}0C(Q) for which Daw exists whenever \a\ = m, define \u\mj, = max ||DQw||p G [0,oo]. \a\=m
| • | m j P is a seminorm when restricted to those u G ^ c W for which Dau G 1^(0) whenever \a\ = m. Lemma 3.4.14 Suppose Q, = (aubi) x ■ • • x (a n ,6 n ) for some - o o < a; < 6; < oo and 1 0 £/iere exists c G (0, oo), depending on m and n only, such that ej\f\j,P < c£ m |/| r o , P + c||/||p
for
f e Cm(il),
0 < e < min&i - au 0 < j < m.
CHAPTER 3. SOBOLEV
134
PROOF
SPACES
Note first that Lemma 3.4.12 implies that
l
e \\D\g\\p < ek\\D?g\\p + Ck\\g\\p for
g e Cfc(ft), 0 < I < k, 1 < i < n.
Choose a multi-index a = ( a i , . . . , a„) with \a\ = j and define fa = a^H fk = Dlk--- D%"f. Note that for 1 < k < n, eak\\fk\\P
eHhWv
=
\-an,
e°k\\Dakkfk+1\\p ||D™-"*+1 A+lllp + C m _ A + 1 ll/fc+ill,,
<
£m-A+1
<
e n , - A + 1 | / l m J , + C , m _ / , 4+1 ||/ fc+ i||p
< em\f\m,P +
Cm-0k+1ePk+1\\fk+i\\P,
where /3 n+ i = 0 and / n + i = / . Hence, if C — maxfc<m Ck, then J\\Daf\\v
= e*||/illp < (1 + C + • • ■ + C " ) £ m | / | m , p + C " + 1 | | / | | p ,
which implies the conclusion of the Lemma.
□
Corollary 3.4.15 For every integer ra > 1 there exists c G (0, oo)7 depending on m and n only, such that l/b> < c|/l J m / ?ll/lli" J ' /ro
for
f e C^(ft), 0 < j < m, 1 < p < oo.
PROOF If / G C^(O) and / = 0 in M n \ft, then / G ^ ( M 7 1 ) ; hence the conclusion follows from Lemma 3.4.14 and (3.15). □ Theorem 3.4.16 Suppose Q = (ai, b\) x • • • x (a n , bn) for some —oo < a; < o^ < oo. Suppose also that 1 < p < oo, / G 1^(11) and i/iai i/iere ernste D a / G 1^(0) whenever \a\ = m. Then there exists D@f G 1^(0) /or all \/3\ < m. Moreover, for some c G (0, oo), depending on m and n only, we have that £j\f\j,p < csm\f\myP-\-c\\f\\p
for
0 < e <minbi-ai,
0<j<m.
i
PROOF Let SI' = (ci,di) x • • • x (cn,dn) for some a; < C{ < d{ < b{. Theo rem 3.4.1 implies the existence of / i , /2, • • • in Co°(IRn) such that /
\fk ~ f\P -> 0,
f
\Dafk - Daf\p -> 0 when \a\ = m as k -> oo.
Lemma 3.4.14 implies that {D^f^} is a Cauchy sequence in Lfffi) for each multi-index (3 with \(3\ < ra. Thus there exists Dpf G 2 / ( 0 ' ) for all \0\ < m and, by Lemma 3.4.14, we have that £j\f\j,P,n'
< c£m\f\m,p,n' +c||/||p > n /
for
0 < e < mind; - Cj, 0 < j < ra.
3.5. DEFINITION
AND BASIC PROPERTIES
OF SOBOLEV SPACES
135
Theorem 3.4.4 implies that D@f exists in O for all \/3\ < m. Letting Q,' -» Q gives the bound. D The above interpolation inequalities are actually true under much weaker assump tions on H. See Adams [1] for more details. An obvious consequence of Theorems 3.4.4 and 3.4.16 is: Corollary 3.4.17 Suppose that f G L/ oc (n) and that Daf exists whenever \a\ = m. Then D ^ / exists whenever \f3\ < m.
3.5
Definition a n d Basic P r o p e r t i e s of Sobolev Spaces
Definition 3.5.1 ofW™*(tl), 1 ^ ( 0 ) , Wm*(Q), Wm(Q), ( v ) m , || ■ \\m,PFor nonnegative integers m and for p G [1, oo], define W^f(£l) to be the set of all u G Lploc{£l) which are such that D a ii exists and Dau G Lpoc(£l) for all multi-indices a with \a\ < m. W£C(Q) = W%f(n). For nonnegative integers m and for p G [1, oo] define Wm,p(fl) to be the set of all u G 1^(0) which are such that Dau exists and Dau G 1^(0.) for all multi-indices a with \a\ < m. Wm(Q) = Wm^{Q). For u,v G Wm(Q), define the inner product (u,v)m by
(u,v)m = Y, |a|
/(DQ^)D%. fi
For u G W m ' p (ft), define the norm \\u\\m,v by
IML,P
= (E|a|<ml|D a u|l5) 1/P
\\u\\m,oo =
max| a |< m ||D%|| 0 0
if 1 < P < °° if
p = oo.
Theorem 3.5.2 Wm*(Si) is a Banach space. Wm{Q) is a Hilbert space. PROOF Clearly, Wm,p(Q) is a normed linear space. If 1*1,1x2, ••• is a Cauchy sequence in Wm>p(Q), then {DaUi} is a Cauchy sequence in 1^(0) for each multi-index a with a < m. Let u denote the 1^(0.) limit of Ui, and let ua denote the LP{Q) limits of DaUi for |a| > 0. Dau = wQ by Theorem 3.4.3. □ Theorem 3.5.3 I / u e Wm*{ti)
and v G C£(ft), then uv G VF m ' p (0) and
HHLj, < 2 m (l + m)n/P||u||m,p||V||m>00.
CHAPTER 3. SOBOLEV
136
PROOF
SPACES
See Theorem 3.4.10.
Lemma 3.5.4 If u G Wm'p(Q),
□
1
0 and J£ is a mollifier, then
lim f \Da(J£*u)-Dau\p
=0
for all compact K C 0. and all multi-indices a for which \a\ < m. PROOF
For all sufficiently small e and for all |a| < m, we have that Da{J£*u)
= Je*(Dau)
mK.
Theorem 3.1.7 implies convergence.
□
The following result shows that Wm,p(Q,), for p < oo, is just the completion of the set of C°°(Q,) functions with finite || • \\mj) norm. This fact is used by some to define Wm*(Q) and, in this case, the notation # m 'P(Q) is often used instead of Wm>p(Q.). Theorem 3.5.5 If p G [l,oo) and m > 0, then C°°(Q) n Wm#(Q)
is dense in
Wm>P(ty. PROOF Choose u G Wm>p(Q,) and e > 0. Let -01,-02, ••• be a partition of unity as in Theorem 3.1.3 (take, for example, r = {Q}). Choose Si G (0,1/i) so that K{ = supp(^i) + B(0,Si) C O . If t G (0,Sj), then Jt * (uipi) G CQ°(Q), supp(J* * (uipi)) C K{. By Lemma 3.5.4 (see also Theorem 3.4.10), \\Jti*(u*Pi)-uil;i\\rniP<e2-i (3.41) for some U G (0,1/i), U < S{. Pick any i e O and let r > 0 be such that B(x,2r) C fi. There exists an integer m > 1/r such that ipi = 0 on B(x,2r) for all i > m. It is easy to see that Jti * (f#i) = 0 on B(x,r) for i > m. Therefore, if oo
0 =^
J t . * (w/>i),
t=l
then in a neighborhood of each point in ft, all but finitely many terms of the sum vanish identically. Thus 0 G C°°(0). The following identity is obviously true pointwise a.e. oo U - 0 = ^
U^i - Jt{ * {Uipi).
t=l
(3.41) implies that u - 0 G W m ' p (Q) and ||u - 0|| m , p < e.
□
3.5. DEFINITION AND BASIC PROPERTIES OF SOBOLEV SPACES
Wm(Rn)
137
can be nicely characterized with the help of the Fourier transform:
Theorem 3.5.6 Suppose f G L2(Rn), m > 0 and g(x) = (1 + \x\2)m/2{Tf)(x) for x G Rn. Then f G Wm(Rn) if and only if g G L2(Rn). Moreover, there exists c G (0, oo), depending only on m and n, such that when f G W™(Rn).
H/IU.2 < H0II2 < c||/|| m , 2 PROOF
Note that for some c G (0,00)
E (**)2 < a + i*i2r = E ( 7 ) E £(*Q)2 *c2 E (*a)2 |a|<m
fc=0
^
2
' |a|=fc
2
2
E i*w)(*)i < iswi < c |a|<m
'
|a|<m
E K^/M^I2-
|a|<m
This and Theorems 3.4.5 and 3.2.6 imply the conclusion.
D
Definition 3.5.7 of W^p{n),W^(Q). Co°(fi) is clearly a subspace of each of the spaces Wm,p(£l). For nonnegative integers m and for p G [l,oo], define W™'P(Q) to be the closure
of cg°(ft) in wm^{n). w$*(n) = w™>2{n).
Clearly, W™'P(Q) is a Banach space with norm || • || mjP and W™(Q) is a Hilbert space with inner product (•, -) m . Theorem 3.1.4 implies that CJ"(0) C W™'P{Q)\ thus, the closure of C^ffi) in Wm>P(fi) is also equal to W™*{Q). This and Theorem 3.5.3 imply that if u G W™*{Q) and v G C£(ft), then tiv G W 0 m ' p (ft). Also, this and Theorem 3.5.5 imply that if p < 00, u G Wm>p{n), v G qrW,
then uv G W0 m ' p (ft).
Except for some special sets fi, we have that Wm*(Q) ± W™,P(Q). One of these special fi is R n . Theorem 3.5.8 Wm^(Rn)
= W™'p(Rn) if p G [l,oo), m > 0.
CHAPTER 3. SOBOLEV
138
SPACES
PROOF Choose u G Wm*{Rn) and let / G C0°°(En) be such that f{x) = 1 for |a;| < 1 and f{x) = 0 for \x\ > 2 (Theorem 3.1.2). Define Ui(x) = u{x)f(x/i) and note that U{ are in W™,p(Rn). Theorem 3.4.10 implies that there exists c such that for all i > 1,
IK - u\\m,V < c _
\a\<mJ\x^
and therefore u G W™ When ft lies between two parallel planes, with a normal v and at a distance d apart, then (3.42) holds. In other words: Theorem 3.5.9 Suppose ft is such that for some v G Rn, with \v\ = 1 and some d G (0, oo), we have that (x — y) • v < d for all x, y in ft. Then ll/llj»
for all f G < ' p ( f t ) , 1 < p < oo.
(3.42)
Choose / G CQ°(Q,) and note that for all x G ft,
f(x) = -
Jo
i/'Vf(x + vt)dt.
Hence, if p < oo and 1/p + 1/q = 1, then \f(x)\p [\f\p=f Jn
jRn
\f\v<
rd
< o?/q / \v ■ Vf(x + ut)\pdt Jo [ W-Vf(x Jo Jw1
+ ut)\Pdxdt = d1+P^ [ Jn
\v-Vf\p,
therefore ||/|| p < d\\u • V / | | p - which is obviously true also for p = oo. Since D \W ' V/Hp < M t h e inequality holds also in W ^ 0 ) (3.42) is called the Ponicare inequality by some (see also Lemma 2.5.1 and the Dirichlet problem in Section 3.7). Others call the inequality (3.43) below the Ponicare inequality. ft, that has properties (i) and (ii) in Theorem 3.5.10 below, is said to be star-shaped with respect to a ball contained in ft. See also Exercise 14. Theorem 3.5.10 Suppose that ft is such that (i) B(x0,ri) (ii) ifyE
C ft C B(x0,r2) B(x0,ri)
for some x0 G W1, 0 < r x < r 2 < oo
and x G ft, then y + t(x — y) G ft for all t G [0,1]
3.5. DEFINITION
and f G C1^),
AND BASIC PROPERTIES
OF SOBOLEV SPACES
p G [1, oo], df/Oxi G LP(Q) forl
139
Then f G LP(il) and (3.43)
for all measurable W C O with measure fi(W) > 0. PROOF
Choose r G (0,ri) and let 5 = 5(a; 0 ,r).
If a; G ft, y G £ , then
fix) = f(y) + Y.j{*i-
vi)^\v + *(* -1/))*;
integrating with respect to y gives
»(B)f(x)= [ /(y)dy+ £>(*), J B
(3.44)
i=i
where *i(x) = f
I (xi-
Vi)^(y
+ t(x - y))dtdy,
x G ft, 1 < * < n.
Let us prove the following inequality: ||Ji||p
3/
for
1 < i < n.
(3.45)
&Ej
(3.45) is obvious if p = oo; so, assume p < oo and note
|ii(*)lp
Jf[ \x-y\
< (jf\x-y\dtdy\ <
(A*(B)(r + r 2 ))P- 1 f
[
JB JO
%L(y + t{x-y)) dtdy
\x - y\ | | £ ( y + t(x - y)) dtdy I axi
f \Ii(x)\'dx <
{fi{B){r + r2))p~l
[ dy [ dt [ JO Jn
dx\x-y
=
1
{»{B){r + r2)y-
-L(y + t{x-y)) OXi
JB
n l
[ dy f dx f dtr - g{x,y,t)\x-yV Ja Jo
JB
f
dxi
(x)
CHAPTER 3. SOBOLEV
140
SPACES
where g(x, y, t) = 1 if x G tQ 4- (1 — t)y and otherwise g(x, y, t) = 0. If x G 0 , y e B, t > 0 and x G £fi + (1 — t)y, then for some z G f&, |# — y| = £|2 — y| < t(r2 + r) and / ' r n - l g { x , y , t ) d t < C , t-^dt 7o J^=fi
< - ( f ^ V n \\x-y\J
Therefore
f \Ii(x)fdx < (fi(B)(r + r2)r'l{r Jn
+ V2)n
«
[ dx [ Jii JB
dy\x-y\l-n
dxi
(x)
and inequality (3.45) follows from
[ \x- y\l-ndy < f JB
\y\l-ndy = n / ^ r 1 " " .
JB(0,r)
In view of (3.44) and (3.45), it is clear that / G 1^(0). Integration of (3.44) over W gives
fi(B) f f = ii(W) [ f+ £
[ *■
Dividing this by //(W) and subtracting it from (3.44) gives
/i(B)
^lip-
Using (3.45) and letting r —>• n proves (3.43).
3.6
□
Imbeddings of Wm^(fi)
In this section we shall study imbeddings of W m ' p ( 0 ) , where m is a positive integer and 1 < p < oo. We shall begin by showing (Theorem 3.6.3) that, under very weak conditions on fi, if mp > n and j > 0, then Wj+m>p(n) C C3B(Q).
3.6. IMBEDDINGS
OF WM>p(Sl)
141
We shall then show (Lemma 3.6.8) that if mp < n, p < q < ^ ^ , q < oo, then Wm^{Q) C L 9 (ft) when ft is a box (possibly the whole W1). This will give us very general local imbed dings (Corollary 3.6.9), imbeddings of W0m'p(ft) (Theorem 3.6.10), as well as Holder continuity of functions in W™'p(ft) for mp > n (Theorem 3.6.12). Under much weaker conditions on ft, it will then be shown (Theorem 3.6.14) that mp < n im plies Wm^(Sl) C L 9 (ft) for p < q < ^ ^ . Sufficient conditions for compactness of the imbedding of Wm*(Sl) into L 9 (ft) are presented in Theorem 3.6.16. Imbeddings of Wrn'p{Q) in general depend on the boundary of ft. On the other hand, W™'p(Sl) is a closure of a set of functions that vanish on the boundary of ft and hence, its imbeddings are, in general, independent of the boundary of ft. The weakest, commonly used assumption on ft for which most imbeddings still exist is called the cone property of ft and is defined as follows: Definition 3.6.1 For a G Mn, a ^ 0, 6 G (0,?r], define cone(a,0) = {x G Rn \ x • a > \x\\a\cos0 and \x\ < |a|}, and observe that if t > — n, then f
\x\*dx = c\a\t+n
Jcone(a,9)
for some c G (0, oo) which depends only on n, t and 9. A nonempty open set Q in Mn, n > 1, is said to have the cone property if there exist h G (0, oo), 0 G (0,ir] such that for every x G fi we can find a G Rn with the properties \a\ = h and x + cone(a, 0) C il. Lemma 3.6.2 Suppose m is a positive integer, p G [l,oo] and either mp > n or m > n. Suppose also that Q has the cone property and let h and 0 be as in the Definition 3.6.1 of the cone property. Then there exists c G (0, oo) such that m
supM p(Q). c depends only on m, n, p and 0. PROOF CI, c 2 , . . . will denote numbers in (0, oo) that are determined by m, p, n and 0 only. Choose u G C m (fi) fl Wm>p(n), x G ft and a G Rn with \a\ = h so that x 4- cone(a,0) C ft. Abbreviate C = cone(a,0) and let c\hn be the volume of the cone C.
CHAPTER 3. SOBOLEV
142
SPACES
If y e C, then x + y - ty G £1 for t € [0,1]. Hence the Taylor formula (1.8) can be written as u(x)=
V
izl|^ya(^au)(a;+y)+m
|a|<m
V-
hll!!l ^
t™-ly«(D«u){x+ty)dt.
|a|=m
Integrating both sides over C gives
u{x)dhn= y
i-^_/Q+m y
|a|<m
^—f-ja,
(3.46)
|a|=m
where f ya(Dau)(x Jc
Ia=
+ y)dy,
\a\ < m
f [ tm-lya(Dau){x + ty)dydt, \a\ = m. Jo Jc Let q e [l,oo] be such that l/p+l/q = 1. Since \ya\ < \y\^, Holder's inequality implies that Ja=
\Ia\
for
\a\=i<m.
(3.47)
When | a | = m the scaling gives
Ja = f
f t-n-1ya{Dau)(x + y)dydt
Jo JtC
= where f(t,y)
[ [ t-n-1f(t1y)ya(Dau)(x
Jo Jc
= 1 if |y| < ht and f{t,y) [ t-n-1f(t,y)dt Jo
+ y)dydt,
= 0 otherwise. Since
= (hn\y\-n-l)/n
for
y e C,
we have \Ja\ n
[ \y\m-n\(Dau)(x Jc and obviously I J«| < hm\u\m^
+ y)\dy-
(3.48)
(3.49)
if p > 1, then (m — n)g + n = g(m — n/p) > 0 and we can apply Holder's inequality to (3.48) to obtain \Ja\
(3.50)
Inserting (3.47) and either (3.49) or (3.50) into (3.46) completes the proof.
□
3.6. IMBEDDINGS
OFWM>p(ty
143
Theorem 3.6.3 Suppose m > I, p G [l,oo] and either mp > n or m>n. the cone property and h and 9 are as in Definition 3.6.1, then Wj+m>p(Q) C CjB{Q) for
If Q, has
j > 0.
Moreover, there exists c G (0, oo), depending only on m, n, p and 0, such that m
Mi,oo < ^hi~n/p\u^p
f°r
u e
wj+m>p{£i), j > o.
PROOF Lemma 3.6.2 and Theorem 3.5.5 imply the conclusions when p < oo. If p = oo, choose r G (1 + n/ra, oo) and let 5 b e a nonempty open ball contained in 0 . Since B has the cone property, Wj+m*(B) C Wj+m>r(B) C C3B(B). Hence, W' + m *(ft) C C'(ft). □ If u G W0m'p(ft) and u = 0 on M n \0, then u G W0m'p(Mn); and since Rn has the cone property, we have Corollary 3.6.4 Suppose j > 0, m > 1, 1 < p < oo and either mp > n or m> Then M^' +m ' p (ft) C CJB(tt) and W^P{Q) C C'(ft).
n.
We shall now begin studying imbeddings of Wm,p into Lq for q < oo. Lemma 3.6.5 IfQ, = (ai,&i) x • ■ • x (a n ,6 n ) /or some —oo < ai
< e'l\\g\\i + |^| l f l
g G Wl>l(Q), 0<e<
for
n—1
mino* - a*. t
1,1
PROOF Suppose p G C^O) n W ^ ) , a» < c» < d» < 6,, e < d» - c», fi' = (ci, di) x ■ • • x (c^ d n ), and let J^ denote integration of £j from Ci to dj. If /i G C 1 ^ , 6] and a < z < 6, then
(6-a)/i(*)= / My)*+ [Z(y-a)ti(y)dyJa
Ja
f
(b-y)h,(y)dy
Jz
Therefore \g(x)\<e-1
[\g\+
f\Dig\
for
z e ft', i = 1,.. .n.
It is easy to see that this implies the conclusion of the Lemma when n = 1; so assume from here on that n > 1 and a = ^^. Note that
towr-(/*)"-(/»*.
144
CHAPTER 3. SOBOLEV
SPACES
where ft = e-l\g\ + \Dig\. Integration of xu using Holder's inequality (1.10) with k= pi = n-l, gives
LvsMWs)'Likewise, integration of r r i , . . . , x^ gives
f a nQ a ^•••^M
r iLji9r0-t\{L - nf(')/ / < ) -n(i/*) ^n(i/<) <(^ ^(£_iiwi+Mi,i) i|s||i+iffii>i)nana;; thus ||^||_s_ < £ _1 ||p||i + |#|i,i and Theorem 3.5.5 implies the conclusion.
□
r IIff l
ll/IIJ-'ll/ll?, ll/ll, < ll/l&-'ll/ll?,
where
I _ I 9 »= =| -pf .j -
^O.OJj (3.51)
If this and the following fact, ^Ali~-e0B#e 0 < £s~-eA^ ++ £zxi~-6B^
, £B >>0,0,£e>>0 ,0,06>€€ [0,1], wwhen h e n 4A >>0 0, [0,1],
(3.52)
are used in Lemma 3.6.5, we obtain Corollary 3.6.6 Suppose fi 0 = (oi,&i) (ai,&i) x • • ■ • xx(a(a„, /orsome some- o- ooo< l - i , Men then (1 ) \\9\\ e\g\1A) llffll,q < ee-- n"^H2\\g\U - ' ( 2 | | S | | 1 ++ eW,i)
for
^(fl), 0< e< < mm mini, a,. 9g 6e W^(Q), & i - a,. i
Lemma 3.6.7 Suppose O = (oi, fcj) x • • • x (a„, &„) /or some - o o < a{ < h < oo. / / l < P < 9 < o o o n r f i > i - i , tten \\h\\ llfcll,q < e-n{r--,\l\\h\\ r--*\l\\h\\PP
+ eq\h\hp)
for
h e W1^*^), ft), 0
<e<mmbi-ai. i
OFWM>p{Q)
3.6. IMBEDDINGS
145
PROOF Assume p > 1. Let s = 1 + q ^ G (1, oo) and pick / G C°°(ft) Pi W 1 * ^ ) . Let g = \f\s and note that g G C 1 ^ ) , \D{g\ < s\f\a-l\Dif\. Let 12' = (ci,di) x • • • x (en,dn) for some ai < Ci < d{ < b{ with e < d{ — C{. Note
IIA
llslkn- < ll/IU'll/ll^1,. If t = q/s, then \ > 1 — ^. Hence Corollary 3.6.6 implies ll/ll^< = hWtfn < e" B ( 1 -« ) (2||/||p, n .||/||;^ ll/IU <
+es\f\1w\\f\\*-i)
e-<-^\2\\f\\Ptn'+es\!\ltPfl,),
and letting Q! -> O and f -> h (Theorem 3.5.5) implies the conclusion.
□
Lemma 3.6.8 Suppose Q = (ai, b\) x • • • x (a n , 6 n ) for some —oo < a; < b{ < oo. / / m>l, 1 ± - ^ , tfien m
l|/i|U < (2 + 9 ) m y"e A : - T l ( p-« ) |/i| J f c , p
/or
fter^fi),
0<e<min6i-ai.
fc=0
PROOF The conclusion of this Lemma is proven in Lemma 3.6.7 when m = 1. Assume that m > 1 and that the conclusion is true for m — 1. Choose 5 so that max
/I
I _ m=!\ < I <
mm
Since m — 1 > 1 , l < p < s < o o and j > ^ —
{I + I Ij. rr2
^ , we have
m—1
ws<(2+gr-i^e':-"("-»)i%,P A;=0 m—1 1
|fc|i, < (2 + q)"*- £ ^""^"^lAU+ij. l
s
fc=0
and hence /i G W > (tt). Since l < s < g < o o , i > j - ^ , Lemma 3.6.7 implies
\\h\\q <
e'n^H2\\h\\a+eq\h\Ua) /
<
m—\
m
(2 + a) m - 1 e- n( "-" ) [2^2ek\h\k,p + q1£Ek\h\Kp \
fc=0
which shows that the conclusion is true also for m.
/c=l
□
CHAPTER 3. SOBOLEV
146
SPACES
Corollary 3.6.9 Suppose j > 0, m > 1, 1 < p < q < oo and ± > ± - ^ .
Then
Theorem 3.6.10 Suppose j > 0, m > 1, 1 < p < q < oo, ^ > ^ - ^ and, when q — oo, it is required also that either mp > n or m > n. Then WQ m ' p (0) C W Q ' 9 ( ^ ) and
l«U, < c\u\]+mtP\\u\\]re for u e wi+m*(si), e = j ^ ( i + 1 - 1 ) , where c G (0, oo) depends only on j , m, n, p and q. PROOF Choose v G CS°(Q) and let v = 0 in Rn\ft. n with E in place of n , imply Hj,g
n(
p J)HJfe+J-iP
Lemmas 3.6.8 and 3.6.2,
for
e>0.
A:=0
Lemma 3.4.14 implies l « U , < c 2 ( e m - , ( p - J ) | t ; | i + m , + e- J '- B ( p-i ) ||t;||p)
for
e > 0;
hence (3.15) implies the bound stated. Since this is true for every j > 0 and C§°(fi) is dense in W$+m*(0), the rest follows. □ Before continuing with our study of imbeddings of Wm,p into Lq, q < oo, let us apply the above result to show the Holder continuity of functions in W^fo) when mp > n. Lemma 3.6.11 If p > n, then there exists c G (0, oo) such that \u(x)-u{y)\
for all x,y G R n , u G C%(Rn).
Suppose u G C^(Mn), x,y G Mn and d = \x - y\. Note that n
„!
u(x) = u(x + z) — Y^ / Zi(Diu)(x + tz)d£. Subtracting from this expression the corresponding expression with y in place of x, and integrating z over the ball B = B(0, d), gives (ti(z) - U(
(3.53)
OFWM>p(n)
3.6. IMBEDDINGS
J x
=
()
147
y2
Zi(Diu)(x +
J
i=l
tz)dtdz
B JO
and co, c i , . . . denote numbers in (0, oo) that depend only on n and p. Using the bound J(x)
=
J2 I
I
B
fei^° J* \J(x)\ < f
t-n-1zi(Diu)(x
+ z)dzdt
r^ditdf^-^U^pdt
JO
c2d1+n-n^\u\ljP,
< as well as 1
=
/ y2(*i-yi) ,B JB~\ i=i n
{DiU){y + z + t(x- y))dtdz
c3Y,\xi-yi\
\i\ < <
JO
c4d1+n-n/p|M|i,p
in (3.53) completes the proof.
□
Theorem 3.6.12 Suppose p E [1, oo] and m is an integer such that 0 < m — ^ < 1. If either m— ^ < 1 or m = 1 or m — n = 1, choose any a E (0, m — ^ ; otherwise, choose any a E (0,1). If 9 = ^ + ^-, then there exists c < oo such that max K ^ u N s ) - (D a U )(y)| < c|x - y | > | $ + m > | , V for all x.yeQ,
all j>0
and all u E
W^m'p(n).
PROOF Since C^M 7 1 ) is dense in W^ +m ' p (ft), it is enough to prove the Theorem for u E C^(Rn) when j = 0. Assume first that m = 1 and let a = a/(l — ^ ) . Lemma 3.6.11 and Theo rem 3.6.10 imply that for all x,y E Mn, |uOr)-u(y)|
<
KxJ-uCyjriu^-iity)!1-0
< c(|x-y|1-?HilP)a^Hf|P||ti||J" = ck-t/riuiJjitiii 1 -
CHAPTER
148
3. SOBOLEV
SPACES
Suppose now that m > 2. Define q = jz^ £ (™> oo] and note that if q = oo, then m — 1 = n. Since p < ^ j - < n < q and | > £ — 2*-jp, we can apply Theorem 3.6.10 to obtain that |w|i,9 < ciM^pHuHp - ^. Lemma 3.6.11 implies \u(x) - u(y)\ < c2\x - y ^ M ^
< c2\x - y\°
CMIMVD
In Lemma 3.6.8 there are severe restrictions on Q; nevertheless, it implies very general local imbeddings. To obtain such results for more general S7, one can proceed in several different ways. One way is to prove extension theorems first, i.e. one needs to find a map E : Wm*(Q) -» Wm>p{Rn) such that for every u G Wm*{Sl) (1) (Eu)(x) = u(x) for x G 0 (2) \\Eu\\m^Rn
< c\\u\\
Such maps E exist under various conditions and thus the conclusions of Lemma 3.6.8 become applicable to more general 0 . We will not use this approach. Instead we will present a simple direct proof of the conclusions of Lemma 3.6.8 for any fi that satisfies the cone condition, provided that h > h ~ ^- This non-sharpness could be removed by simply using the Hardy-Littlewood-Sobolev inequality instead of Young's inequality to prove (3.55) in the critical case where ± = ^ — ^ > 0, p > 1. Lemma 3.6.13 Suppose Q, has the cone property and let h and 0 be as in Defini tion 3.6.1 of the cone property. If m>l, 1 < p < q < oo and ^ > £ — ^ , then there exists c < oo, depending only on m, n, p, q and 9, such that m
\\u\\q
for
U
GCm(l])nr^).
i=0
PROOF If p = q, the statement is obvious; if q = oo, then the assertion follows from Lemma 3.6.2. So, assume p < q < oo. ci,C2,... will denote numbers in (0, oo) that are determined by m, n, p, g and 6 only. For j = (ju...Jn)eZn,
define
Box(j) = {x G Rn | ji < Xi/h <ji + l for all t = 1 , . . . , n} Box(j)* = {x e Rn | ji - 1 < Xi/h < ji + 2 for all i = 1 , . . . ,n} and observe that Box(j)* is a union of 3 n disjoint boxes, Box(j a ), that are adjacent to Box(j). Let {Vi,l^,...} denotes the collection (possibly finite) of those sets QnBox(j), j G Z n , which are not empty. For each Vi there exists a unique j G Zn such that Vi = ftnBax(j), define V? = OnBox(j)*. The following facts will be needed later on:
3.6. IMBEDDINGS
OF WM>p{n)
149
(Fl) Vi H Vj is empty if i^ j ; Q = UV{ (F2) if x G Vu y G Q, and \x - y\ < h, then y G V*
(F3) i f / e L 1 ^ ) , / > 0 , then
E /
/<3n//-
Choose any x £ Q. There exists a unique Vi such that x G Vi. Let a e Rn be such that \a\ = h and a; + cone(a, 9) C ft. Abbreviate C = cone(a, 9) and let c\hn be the volume of C. Observe that x + C C V-*, by (F2). In exactly the same way as in the proof of Lemma 3.6.2 we obtain that
u(x)cihn=
Y,
^ 7 ^ a +m E
|a|<m
^T-Ja,
|a|=m
where \Ia\<
[ \y\lal\(Dau)(x Jc
+ y)\dy,
\a\ < m
\Ja\
[ \y\m-n\(Dau)(x + y)\dy, \a\ = m. Jc The change of variable x + y -> y and the fact that x + C CV* give |tx(x)| < c2 E
h~nFa(x)+C2
|a|
J2
(3.54)
|a|=ra
where Fa(x) = [ \x - y\W\(Dau)(y)\dy, Jvt* Ga(x) = [
\x - y\m-n\(D«u)(y)\dy,
x G Vu \a\ < m x G VJ, \a\ = m.
Observe that Fa(x), Ga(x) are uniquely defined for each i G f i and that (3.54) holds for every x G ft. Fix any multi-index a such that \a\ < m. Let t = \a\ if |a| < m and let t = m — n if |a| = m. Define / on all of Q by f(x)=
[
|x-y|«|(D°u)(y)|dy
for
* G V*.
Hence, / = Fa if |a| < m and / = G a if \a\ = m. f equals k * (xi|I> a w|) in Vi, where x» is the characteristic function of the set V*, k(y) = \y\* when |y| < 2nh and 0 in the rest of Rn. Define \ = 1 - \ + \ G [±, 1) and note that
CHAPTER 3. SOBOLEV
150
SPACES
(m-n)s + n> 0; hence, ts + n > 0 and therefore k 6 L s (M n ), ||fc||A < c 3 /i' + n / s . Young's inequality (page 112) implies
(X.
< c3ht+7 (f
/')'
\Dau\A ' .
(3.55)
Hence / fq
\D«u\p,
[
JVi
Jv?
and (Fl), (F3) imply f fq < clhqt+!f\\Dau\\l-pY Jn
[ \Dau\p < 3n4hqt+rf\\Dau\\q-p i Jyi 11/11,
[ \Dau\p J^
Therefore h-n\\Fa\\q < c^^-p^u^p
if \a\ = i < m
||Gf a ||,
^
\a\=m
and if these bounds are used in (3.54), the assertion of the Lemma follows.
□
Theorem 3.6.14 Suppose O has the cone property and let h, 0 be as in the Defini tion 3.6.1 of the cone property. If j > 0, ra > 1, 1 < p < q < oo and h > h — 1^, then Wi+rn'p(Q) C W^q{Q) and there exists c < oo, depending only on m, n, p, q and 0, such that m
M^Kc^^^Mi+j*
for uewi+m>p(n).
i=0
PROOF If q = oo, then the conclusions follow from Theorem 3.6.3 and oth erwise from Lemma 3.6.13 and Theorem 3.5.5. □ We shall now present sufficient conditions for compactness of the imbedding of Wm,p into Lq. We need first to generalize the Arzela-Ascoli Theorem to IP spaces. Let Th denote the translation operator, (Thu)(x) = u{x + h)
for
x, h e Rn.
Theorem 3.6.15 Suppose that {tif}g 1 is a bounded sequence in LP(Rn), 1 < p < oo, such that for each £ > 0 there exists S > 0 so that \\ThUi - Ui\\p < e for all
\h\ < 6, i = 1,2,....
3.6. IMBEDDINGS
OFWM>p{Sl)
151
Then there exist integers n\ < ri2 < ■ • • and v G ^(W1)
such that
lim / \uni(x) — v(x)\pdx = 0 for all r G (0,oo). *->°°./|a:|
For every A > 0, i G N we have that J\*UiE
C°°(R n ) and
||^A*Wz||oo < ||J r A|| 9 SUp||Uj|| p 3
\\ThJ\*Ui~
J\*Ui\\oo
= \\J\ * {ThUi - u^Woo < llJAllgSUpHT^tij
-Uj\\p
3
for all / i € l n , where 1/q = 1 — I/p. Thus, we can repeatedly apply the ArzelaAscoli Theorem 1.1.5 to obtain infinite sets of integers N D A\ D A 1, lim i-+oo,ieAk
sup \J\/k * Ui — Vk\ = 0 for all
r G (0, oo).
B(0,r)
Let no = 1 and pick nk G A such that n* > n^_i for k > 1. Note that k
lim sup \J\/k * Un — Vk\ = 0 for all
r G (0, oo), k > 1.
(3.56)
*->°° fl(0,r)
We claim that {uni} is a Cauchy sequence in 2^(5(0, r)) for any r G (0, oo). Fix an r G (0, oo) and pick e > 0. Since for all w G L^R* 1 ), A > 0, (JA*u)(a;)-u(a;) \(Jx*u)(x)-u(x)\p
= <
/
J\(y)(u{x
- y) -
u(x))dy
Jx(y)\u(x-y)-u(x)\pdy
[ JB(O,\)
||JA*ti-ti||J
<
Jx(y)\\T-yu-u\\pdy
/ /5(0,A)
\\Jx*u-u\\p
<
sup \\Tyu - u\\p, |y|
we can choose k so that \\J\/k *v>i — v>i\\p < e/6
for all
i G N.
(3.56) implies that there exists N such that for all i > N, \\J\/k * un{ ~ vk\\PlB{o,r) < ^(B(Q^r))l/p
SU
P IJ\/k * Urn - vk\ < e/3.
B(0,r)
CHAPTER 3. SOBOLEV
152
SPACES
Therefore, if i,j > N, then \\UTH ~ unj llp,B(0,r) \\UTH ~ J\jk
=
* Uni + J\/k
<
* Uni ~Vk+Vk-
* unj
J\/k
+ Jl/fe * W nj - Unj || p ,B(0,r)
e/6 + e/3 + e/3 + e/6 = e,
which proves the claim. 1^(5(0, A;)), A; € N, limits of {wni} define a function v almost everywhere in n E n and, since |M|P)£(o,fc) < swPj \\uj\\p < °°5 w e n a v e t n a t u ^ I^(^n)The following Theorem is our main result on compactness of imbeddings. It says, that when an imbedding exists, then it is also compact except possibly at the critical values of parameters. Theorem 3.6.16 Suppose Q is bounded and that it has the cone property. If j > 0, m > 1, 1 < p < oo, 1 < q < oo, ^ > ^ — ^ and {wj}g 1 is a bounded sequence in Wi+m,p(Q), then there exist integers n\ < ri2 < • • • and v G W^q(Q) such that lim \\uni -v\\ja
= 0.
i->oo
PROOF
Assume first that j = 0.
Choose r G (g, oo) fl (p, oo) such that f > i — ^ - Theorem 3.6.14 implies that ||u|| r < ci||w|| mjP for all u G Wm*{Sl). Let M = sup^ |K|| m , P and define m = 0 in Rn\n. Note that |K||i, R n < | | ^ | | r / x ( ^ ) 1 - 1 / r < c1M^{Q){-l/r. We claim that sup^ \\Thui—ui\\i^n. —> 0 as \h\ ->• 0, where T/j is the translation operator. Pick e > 0. Choose a compact If C O such that 4ciM/i(fi\A') 1 - 1 / r < e , and pick d G (0, dist(i
/
/Rn\o;
| ^ ( x + h) - Ui(x)\dx < e/2
for
i G N, |/i| < d.
(3.57)
For / G C°°(ft) fl Wm*(Q) and \h\ < d we have that
[ \f(x + h)-f(x)\dx JijJ
= [\[ JUJ \Jo
Ythi{Dif)(x +fa)dt ■ ->
dx
3.7. ELLIPTIC
PROBLEMS
153
< [ [
TMDif^x^dxdt
/O Jht+u JO Jht+u) • i=1!
<
n|/l|||/||1,pM(Q)1-1/P
by using Holder's inequality twice. Hence, Theorem 3.5.5 implies that
/ i \Ui(x
+ h) -
Ui(x)\dx
< n\h\Mn{®)l~l,p
for
i 6 N, \h\ < d.
This and (3.57) imply \ui{x + h) - Ui(x)\dx < e/2 + n|/i|M//(ft) 1 - 1 / p
/
for
i e N, \h\ < d,
n
JR
which proves the claim. Theorem 3.6.15 implies that there exist n\ < ri2 < • • • and v 6 L 1 ^ ) such that uni converge in L1(Q) to v. (3.51) implies that {uni} is also a Cauchy sequence in L 9 (fi). This proves the Theorem in the case j = 0. Since {DaUi} is a bounded sequence in Wm,p(£l) for all \a\ < j , we can extract subsequences {Dauni} which converge in Lq(Q,) for all \a\ < j - implying convergence of {uni} in W^q{Q). □ Corollary 3.6.17 Suppose that Q, is bounded, ra > 1, 1 < p < oo, 1 < q < oo and o ^ \ ~ Xn' Then the identity map from W™,P(Q) to Lq(Q,) is compact.
3.7
Elliptic Problems
A version of the Dirichlet problem can be stated as follows: for a given / , find u such that (Aw) (a;) = -f{x) for i G f i u(x) = 0 for
x e dtt.
Green's identity formally implies / Auv = -Y^ / DiU~Div+ / TT-V. Jn jri Jn Jdn on Thus, if v vanishes on dft just like u does, then Yli=i fn DiuDiv suggests the following definition of a sectorial form: n
= fn fv. This
r
$(u,v) = Y\ / ViuDiV ~1 Jn
for
u,v€V
(3.58)
CHAPTER 3. SOBOLEV
154
SPACES
where V = WQ(Q). If % = £ 2 (fi), then assumptions H I , H2, H3 of the section on Sectorial Forms are satisfied. The Dirichlet problem can be reformulated as: for given / E U find uGV such that #(u, v) = (/, v) for all v £V;
(3.59)
or equivalently, find u G T)(A) such that An = / , where A is the operator associated with #. The sectorial form 5 and the operator A associated with it are defined without any restrictions on Q - any nonempty open set in E n will do. Here are some of the properties of A: (1) A is self-adjoint (Corollary 2.8.5). (2) Let X(Q) be the largest of real numbers A for which V
f \DiU\2 > X f \u\2
for all
ueW^(Q).
Theorem 2.8.7 implies that A(ft) G a(A) C [A(ft),oo). If u G W$(Q) and u = 0 outside fi, then u G WQ(Q!) for every open set 0! D O which implies that A(ft) > A(ft')(3) If Q has a finite thickness d in some direction, then the Ponicare inequality (3.42) implies that X(Q) > d~2\ hence, the Dirichlet problem (3.59) has a solution for every f GW, and, by Theorem 2.8.9, the solution is the minimizer of
$(v,v)-2Re{f,v),
veV.
(4) If n is bounded, then A has compact resolvent which implies that X(Q) is the smallest eigenvalue of A (see Corollary 3.6.17 and Theorems 2.8.2, 2.6.8). It is easy to see that the operator A can also be represented as follows: u G T){A) iff u G wf(Q) and there exists / G L 2 (0) such that (tx, A) = - ( / , ) for all G Cg°(ft); ^4w = / . This representation is usually abbreviated as
Aw = - A n for tx G V(A).
(3.60)
One can show that when the boundary of Q, is nice enough, then T)(A) = WQ(Q) D W2(Q,)] however, it can happen that ?)(A) (£ VF 2 (0), see Exercise 13. Regularity of solutions of elliptic problems is investigated in Section 3.8 where it is shown in particular that if u G T>(A), then u G W20J$l) (Example 3.8.4). The one dimensional case is discussed in Examples 2.8.1, 2.8.3 and 2.8.8. One can attempt to solve the problem Aw = — h in Q, and w — g on dfi, by first choosing a smooth enough v defined in Q, such that v = g on d£l and then declaring the solution to be w = v + u where u satisfies (3.59) with / = h + Av.
3.7. ELLIPTIC
PROBLEMS
155
The Neumann problem can be stated as follows: for a given / , find u such that (Au)(x) = -f(x)
for
xetl
du for xedn. on The variational formulation of the problem can be stated again by (3.59) where 5 is also given by (3.58), the only difference being that V = W1^) in this case. In this formulation, Q can again be any nonempty open set in Mn; however, du/dn does not necessarily make sense on the boundary - more assumptions are needed about the boundary of Q, to show that the the solution of (3.59) actually satisfies du/dn = 0. Here are some properties of operator A associated with this $:
— (x)=0
(1) A is self-adjoint (Corollary 2.8.5). (2) <J(A) C [0,oo) (Theorem 2.8.7). (3) if Q, is bounded and has the cone property, then A has compact resolvent (The orems 3.6.16, 2.8.2), see also Theorems 2.6.8 and 1.7.16. (4) if Q is bounded, connected and has the cone property, then N(A) consists of constant functions only (Theorem 3.4.8). Hence, the Neumann problem (3.59) has a solution iff fn f = 0 (Theorem 2.2.8). Other choices of a closed subspace of W 1 (0) for V correspond to other boundary conditions. For some boundary conditions, the sectorial form 5 must also be modified (see, for example, (2.58)). Let us now turn to the general 2nd order elliptic differential operator, n
n
o
/
n
\
n
Q
Green's identity suggests to consider the form, n
$(u,v) — S^y^
n
p.
n
»
p
/ dijDiuDjV 4- Y^ / diDiiiv + / a$uv
for
u,v EV,
(3.61)
where V is a closed subspace of W 1 (fi). We assume that a^-, ai are bounded, complex valued, measurable functions on Q, such that Ima^ = I m a ^ and that there exists S > 0 such that n
n
Re Y, Ylaij(x)tej 2=1 j = l
^ S\Z\2
for a11
?e
RU x e a
'
( 3 - 62 )
CHAPTER 3. SOBOLEV
156
SPACES
(3.62) is known as the strong ellipticity condition. (3.62) and I m a ^ = imply n
Imaji
n
> S\z\2
Re^2J2aij(x)ziZj
for
z e Cn,x e H,
i=i j=i
and therefore Re
E E / OijOiuD^^Sj^ [ \DiU\2 = S(\\u\\l2 - \\ug) Reff(ti,t*) > S(\\u\\l2 - \\u\\l) - c||ti|| 1|2 ||ti|| 2 ,
where c = \\ J27=o la»|2lloo • Hence, for any Ms e (0,6), there exists a eR such that Re£(u,ti) > M 3 |M|? >2 + a||u\\%
for all
ueV.
(3.63)
For elliptic problems, the inequality of form (3.63), with M3 > 0, is called Garding's inequality. If % = L2(Q,), then assumptions H I , H2, H3 of the section on Sectorial Forms are again satisfied. In particular, when Q, is bounded and has the cone property, we can again apply the Fredholm alternative: problem (3.59), with 5 given by (3.61), has a solution iff (/, v) = 0 for all v 6 V such that $(g, v) = 0 for all g 6 V. Consider now the biharmonic operator, Lu = A2w
with u = — = 0 on dQ. on
Green's identity suggests we define ff(u,u) = / AuAv Jn Hue
for
u,veW${n).
(3.64)
Cg°(ft), let u = 0 in R n \ft and note that
ff(u,u) =
f
K|4|fi(0|2de 22
> \ l o. + \z\ ) m)\2d£- f \m\2dt * JRn
JRn
~ IHli;
(Theorem 3.5.6)
> \\M\h and since CQ°(Q,) is dense in WQ (fi), we obtain Garding's inequality
5(11,11) >5lHll i 2 -|H|l for ueW*(Si). Let A be the operator associated with this #. A is self-adjoint and cr(A) C [0,00) as before. If H has finite thickness d in the direction of one coordinate axis, then two
3.7. ELLIPTIC
PROBLEMS
157
applications of Hormander's Lemma 2.5.1 give that $(u,u) > d~*\\u\\2 for u G C™(Q) and hence for all u G WQ(Q); therefore a(A) C [cT4, oo). Corollary 3.6.17 implies that A~l is compact when Q is bounded. If the boundary condition u = du/dn = 0, of the biharmonic operator A 2 , is replaced by u = Au = 0 on dQ,, then its natural proper definition in H — L2(fJ) becomes A2 where A is given by (3.60). Note that if V = T>(A) and if for u,v eV [u,v] = (Au,Av) + (u,v),
\u\ = [u,u]ll2
${u,v) = (Au,Av), 2
then A is the operator associated with this J . See also Exercise 19 of Chapter 2. Observe also that it can happen that V <£. W2(Q) in this case (Exercise 13). The general strongly elliptic operator of order 2m is given by Lu
= E E (-l)WD^(aa0Dau) \a\<m\P\<m
provided there exists S > 0 such that
Re Y,
S
a
<*p(x)£a£0 > s\Z\2m
f o r a11
£ G Mn,z G 0.
(3.65)
|a|=ra \/3\=m
Define $(u,v)= ^
Yl
|a|<m|/?|
f aapDauD^ for u , v 6 r ( f i ) .
(3.66)
n
It is easy to see that |5(w,v)| < M2|M|m,2|M|m,2 f° r u,v £ VF m (0) provided only that aap are bounded; however, proof of Garding's inequality can get rather compli cated when m > 1. When assumptions of the following Theorem 3.7.1 are satisfied, assumptions H I , H2, H3 of the section on Sectorial Forms are again satisfied, with H = L2(Q) and V = W™(Q). Moreover, Corollary 3.6.17 implies compactness of the identity map from V —> %. Hence, the operator associated with 5 has compact resolvent and the Predholm alternative applies as above. For regularity of solutions see Theorem 3.8.2. Theorem 3.7.1 Suppose that Q, is bounded and 5 is given by (3.66), where (a) ra > 1 and aap G L°°(Q) for all \a\ < m, \/3\ < m (b) aap G C(Q) when \a\ = \/3\ = m (c) strong ellipticity condition (3.65) is assumed to hold for some S > 0. Then for every Ms G (0,5), there exists a G R such that Re3T(«,«) > Afallull^a +o||t*||i
for all ueW^(U).
(3.67)
CHAPTER 3. SOBOLEV
158
SPACES
PROOF Choose u G C^{0). We shall use R\, i?2, • •. to denote various terms of $(u, u) which can be estimated as |ft|
30u,u)
= ]T
f o.apDauDPu + R1.
E
\a\=m\/3\=m
n
Define 6 = (8 - M 3 )/3, e = 0(E| tt |=m i ) " 1 \aa/3(x) - aap(y)\ <e
a n d let
A* > °
if \x - y\ < 2/z, x,y eft,
be such
that
\a\ = \/3\ = m.
Note that ft C B{x\,ii) U • • • U B(xk,fj,) for some Xj G ft. Choose real valued (j)j G Co°(Rn) such that s u p p ^ ) C B{XJ,2/J,) and j(x) > 0 for re G B(XJ,H), j = 1 , . . . , k. Note that = \ H h (f)\ > 0 in ft. Hence, ^ = (j>j(t>-l/2 G C £ ( n ) n C°°(ft) and Vj = utpj G Co°(ft) for j = 1 , . . . , k. Since V>? H ^D«u
= Davj - £
(a)
4- ty\ — 1 in ft and
(DPu)D«-%,
it follows that
* ff(u,u)-fli
=
r
E
E
E
a«ptfDauDPu
/
Jft
j=l\a\=m\0\=m
*
=
/»
E
E
aa0DavjD^vj
E
*
S(u,u) -Ri-R2
= E
+ R2
\a\=m\/3\=mJQl
3=1
r
E
a
E
^ ( ^ j ) / # % £^.7 + G,
i = l |a|=m |/3|=m
^n
where *
G
=
EE
r
E
(aa(3 - a^ixj^D^j
j=l\a\=m\(3\=mJB^^nn
\°\ z e E E E J^VJWD^ j=l\a\=m\0\=mJn
D^VJ
3.7. ELLIPTIC
PROBLEMS
159
'Jl j=l |a|=m|0|=rri n
j'=l |a|=m n
Let ^j = 0 on JR \Q and note that
Z(U,U)-RX-R2-G
E <W(^K Q ^(OI 2 *;
= E/„E j=l"
/Rn
|a|=m|/?|=m
k
u,u) - R - R - G) > > syi sir f \em\vj(OM ite(ff(u,«)-iii-ife-G) J« r
2
j=1
E /„ E n>M)M EI^I2
= *E/ j=l
Jn
\a\=m
, / n
|a|=m
k j=l
= (S-0) [ Y, \Dau\2 + Rs |a|=m
= ( * - *)(IMlS,,2 -IHI 2 „-i, 2 )+i?3. Therefore Re$(u,u)
> (S - flJdlull^ - IMI*,-i, 2 ) - C 4 ||«||m-l,2||«|U,2 >
(S-2e)\\ufmt2-c5\\u\\l_lt2.
Since x r a _ 1 < r x m + r 1 _ m for all z > 0 , T > 0, Theorem 3.5.6 implies that N l l - l ^ ^ H ^ +T^Mli and taking r , such that rc^c2 < 0, implies (3.67).
for
r>0, □
CHAPTER 3. SOBOLEV
160
3.8
SPACES
Regularity of Weak Solutions
In this section it will be assumed that m is a positive integer, aap G C(ft) for all multi-indices a, j3 with \a\ < m and |/3| < ra, and that
Re Y,
E
a
<*p(x)Za+P > °
for
* G ft, £ <E M n \{0}.
(3.68)
|a|=m |/3|=m
Observe that this implies that for each compact, K C ft there exists £ > 0 such that R
e
a a / j(x)e a + / ? > <*|£|2m
^
for
x e K, £ e Rn.
\a\=m \(3\=m
For u G W^ c (ft), 0 G C0°°(ft), define
£ J aa0DauD^.
3(11,0= £
\a\<m\(3\<m
(3.69)
n
Lemma 3.8.1 Suppose that 1 < j <m, aap G C J (ft) /or |a| < m7 |/?| < m and that u G Wj™ (ft) is swc/i £/ia£ /or each compact set K there exists CK < °o such that Ifffa, 0)| < C^||0|| m -j,2
for all <j> G (70°°(ft) with supp() C if.
PROOF Assume first that j = 1. Let If be any nonempty compact subset of ft and let d > 0 be such that (5 4 C ft, where 0{ = K + 5(0, id), 1 < z < 4. Let / G C§°(Mn) be such that f{x) = 1 for x G Oi and /(a;) = 0 for x 0 02, see Theorem 3.1.2. Define v = fu. Using notation vy(x) = v(x + y), note that vy G ^ 0 m ( ^ 3 ) for all |y| < d. Pick any 1 < i < n, - d < h < d, h / 0 and 0 G C§°((!?3). Observe that S^v G ^ 0 m ( ^ 3 ) , where 6% is defined by (3.38). Let RUR2,... denote various terms of ^(S^v^c/)) that can be bounded by Cfc||0||m,2 for some Ck that do not depend on h and (j). Note that
WN,0= E E / ^(^Da(/W))i)^. |a|<m|/?|
03
Theorems 3.4.10 and 3.4.6 imply that
5(^,0) = E
E
/
a^(*?(/D°ti))^ + ili.
3.8. REGULARITY OF WEAK SOLUTIONS
161
Since ^{wg) = g5^w + whei5^g, we have
3^,0
=
E
/ ( ^ ( ^ / D a u ) - ( / D ^ ) / i e ^ ^ ) ^ + ,R1
E
M<m|j8|<7n =
°3
E E / SHoapfV^Dfy \a\<m\0\<mJ°3
= ~ E
E
= - E
E
« a ^ / D a u 5 - / l ^ + JR2
/
|a|<m|/3|
+ R*
°
2
«afPaufDHrh4> + Bz
/
|a|<m|/3|<m , / 0 2
= " E
E
H<m|/?|
|ff(*N,0)|
<
aa/9DaU^(/^)+/23
/ °
2
^J|/J-VlU-l,2+C3||0|| m ,2|| m ,2,
and since C§°(03) is dense in Wom(<93), | f f ( * N , ^ ) | 0, a € M we have a||^||i + M3||^||^2
for
\a\ < m,
JK
Theorem 3.4.7 implies that D^u exists and DjDQu 6 Lf0C(Q) for all \a\ < m. This completes the proof in the case j = 1. Assume now that 2 < j < m and that the assertion of the Lemma is true for j — I. Hence we may assume that u e W^3~ (0). Choose any compact set K C ft, (j) e Co°(0) with supp(0) C K and 1 < i < n. Let ci,c 2 ,... denote constants that do not depend on . Note that $(DiU,) =
E
E
[ aa0(DiDau)DP(t>
Jn
\a\<m\/3\<m
= -$(u,Di(i>)-
E
E E«f>>
\a-\<m\P\<™
CHAPTER 3. SOBOLEV
162
SPACES
where Ea/3= [ Diaaf,DauD0. Jn If \{3\ <mj + 1, then obviously \Ea3\ < ci|M| m _j+i >2 . If \0\ > rn - j + 1, then ft = 7 + /i where \y\ = m — j + 1 and |/J| < j — 1; hence D^(Aaa/jDQu)i)^
Ea3 = (-l)W / and \Eap\ < C2||0|| m -j+i,2 a l s o \d(0iU,4)\
m
this case. Therefore
< CK\\Di(j)\\m-ja
4-c 3 ||^|| m -j+i,2 < aWW m—j+1,2
and the induction assumption implies that Djix G W^3~
(ft).
□
T h e o r e m 3.8.2 Suppose that m > I, k > 0, aag G Cm+k(£l) for \a\ < m and \P\ < m satisfy (3.68), and that u G Wj£.(Sl) and f G W£c(f2) are such that E
E
ThenueW?™+k{n)
[ ^pOauDp(/>
= j f(f>
for all 0 G C0°°(O).
and
J2 ^(- 1 ) l / ? l D '(^ D ^) = /|a|
PROOF Lemma 3.8.1 implies the assertion when k = 0. Assume that k > 1 and that the assertion is true for k - 1. Note that u G W^ + f e ~ 1 (fi) and that for every 1 < i < n, (j) G Co°(Q), we have
£
f aaPDauD^Di(t>=- f ^DJ
E
\a\<m\3\<mJn
DiuD^ H<m|0|<m,/"
E
E
Jn
=
j Wif^
a
/ aafiD
=
where 5 = 0,/- ^
/
E
E
^01
^(-l)l«iy(Aao/jDat
\a\<m\3\<m 1
f
|a|<m |jS|<m"
and, since p G W ^ T ^ ) , we have that D;u G W ^ * " 1 ^ ) -
Dia^D^uD^
3.9.
EXERCISES
163
Theorem 3.8.3 Suppose that m > 1, k > 0, ba G C2m+k(Sl) for \a\ < 2m and M z ) £ Q > 0 for all x G ft, £ G Mn
Re C E |a|=2m
/or some ( G C . Suppose also that u G Z^oc(Q) and / G W^C(H) are suc/i that f u ] T baDa = I f(f> for all 0 G Cg°(ft). T/>en ti G ^ ^ ( Q ) and £ , a , < 2 m ( - l ) l " ' D « ( o ^ )
=
/.
PROOF Pick any nonempty compact subset K of Q. and let d > 0 be such that O C ft where 0 = X + 5(0,d). w —>• (w,w) = f0wu is a bounded linear functional on VFm((9); hence the Riesz Lemma 2.2.4 implies that there exists v G Wm(0) such that (w,u) — (w,v)m for all w G W r m (0) and, in particular, V
/ DavDa(/)=
f ucj) for all 0 G C0°°(O).
|a|
Theorem 3.8.2 implies that v G W£™(0) and u = Y,\a\<m(-1)la]D'2av-
E
Hence
E / ( - l ) H + m C ^ D 2 a t ; ^ = / ( - i r C / 0 for 0GCO°°(O)
and Theorem 3.8.2 implies that v G W ^ + * ( 0 ) ;
hence MG w
io™ + *( 0 )-
D
EXAMPLE 3.8.4 If / G W£ c (n), A: > 0 and u G L2loc(Q) are such that [ uAp=
[ ftp for all y> G Cg°(fi),
then Theorem 3.8.3 implies that u G W£+fc(ft) and Au = f. In particular, if / G C°°(ft), then u G C°°(n) by Corollary 3.6.4.
3.9
Exercises
1. Supply the details for the following alternative proof of completeness of normal ized Hermite functions in L2(M). If (/, n) = 0 for all Hermite functions (f)n, then the Fourier transform of the function f(x)e~x I2 is 0 and hence / = 0.
CHAPTER 3. SOBOLEV
164
SPACES
2. The electric and magnetic fields E(x,t),B(x,t) E C 3 satisfy Maxwell's equa tions, Bt = -V xE, Et = V xB, in R 3 for t > 0. Solve the equations for given £?(-,0),£(-,0) E S 3 . 3. Derive (3.11) from (3.6). 4. Prove (3.16). 5. Show that if / E <S and t e E , then g,h E <S, where 0(x) = /(x) cos |z|t,
h(x) = / ( g ) S " L f •
6. Suppose that 1 < p < oo and —oo < a < 6 < oo. Show that / E W0'p(a,b) iff / E ;4C[a, 6], / ' E Z?(a, 6) and / ( a ) = /(&) = 0. 7. Suppose u E L[ oc (fi), {/i, /2, • • •} C Lj-oc(tt), a is a multi-index, D a /j exists for i > 1, c = supj ||D a /i|| p < oo for some p E (1, oo] and lim [(u-fi)ip
= 0 for all
ip E C0°°(ft).
Show that Daw exists and ||Daix||p < c. Show also that the restriction p > 1 is necessary. 8. Observe that 0, in the proof of Theorem 3.5.5, is a sum of CQ?(Q.) functions. Is <j> E
W™*(Q)1
9. Is it true that W™,p(Rn) = Wm>p(Rn) also when m > 0 and p = oo? 10. Show (3.51). 11. Define u(x) = log(log(4/|x|)) for x E 5(0,1) C Rn. Show that uEVr^(5(0,l))\L°°(5(0,l)) when p > 1 and rap = n. (See Lemma 3.6.2.) 12. Suppose ra E N, p E [1, oo] and 0 < r a - ^ < / z < c r < l . Define u(x) = \x\v for x E 5(0,1) C Mn. Show that u€Wm*(B{0,l))\Ctr(B(0,l),C). (See Theorem 3.6.12.)
3.9.
EXERCISES
165
13. Let ft C E 2 be given in polar coordinates (r, (/>) by 0 < r < 1, 0 < <> / < TT//3 where 1/2 < /? < 1. Define w = (1 - r)r? sin/? in ft. Show that u € D(A), where A is given by (3.60). Also show that u g W2(Q). 14. Let ft be a nonempty, bounded and connected set in E n with the cone property. Apply Theorem 2.6.8 to the Neumann problem to show that there exists c < oo such that
lk-^/^l|2^cEnD^ii2 for feWlV)II
M^J Jn h
fr[
What role does this c play in the Neumann problem? Note that this removes the star-shaped condition on ft in the Ponicare inequality (3.43). 15. Show that Garding's inequality (3.63) still holds for J given by (3.61) when V = Wo (ft) and the condition Ima^ = I m a ^ is replaced by Im(a^ — aji) G
Chapter 4
Semigroups of Linear Operators 4.1
Introduction
We shall study problems related to the abstract differential equation of the form u' + Au = 0, where A is a linear operator in a Banach space X and u' denotes the derivative of u : [0, oo) —► X in the Banach space, i.e. lim -r{u{t + h
h)-u(t))-u'{t)
0.
/i->0
Assuming that for given u(0) the equation has a unique solution on [0, oo) implies that there exist linear operators Q{t), for t > 0, such that u(t) = Q{t)u(0) and Q(t)Q(s) = Q(t + s). Formally, Q(t) = e~At. Assuming continuous dependence on initial conditions gives us that Q(t) should be a bounded linear map from its domain into X. In applications it can be usually arranged so that the domain of Q(t) is the whole X. Clearly, the map t —> Q(t) has to have some continuity properties. This, and some technical considerations, suggest the following definitions. Definition 4.1.1 A family of bounded linear operators {Q{t)}t>o on a Banach space X is called a strongly continuous semigroup (or CQ semigroup) if (a) Q(0) = 1 (identity map) (b) Q(t)Q(s) = Q(t + s) for all* > 0,a > 0 (c) limt_K)+ \\Q(t)x -x\\=Q
for all x E X. 167
CHAPTER
168
4. SEMIGROUPS
OF LINEAR
OPERATORS
D e f i n i t i o n 4 . 1 . 2 Let {Q(t)}t>o b e a strongly continuous semigroup of operators on a B a n a c h space X. Define T)(A) to be t h e set of all x E X for which there exists y e X such t h a t lim \hx-Q(t)x)-y\\ :0; t->-o+ t for such x a n d y define Ax = y. T h e linear operator, —A, is called t h e g e n e r a t o r (or t h e i n f i n i t e s i m a l g e n e r a t o r ) of the semigroup. EXAMPLE 4.1.3 Let the Banach space X be CU{R). For t > 0 define Q(t) G
= f(x -t)
for
feX,xe
R.
One can easily verify that {Q{t)}t>o is a strongly continuous (but not continuous, Exercise 1) semigroup of operators on X. Let —A be its generator. If / G T>(A) and e(t) =f~
Q{t)f
- Af
for
t>0,
then limt->.o+ \\e(i)\\ = 0 and, since for all i e R , f(x-t)-f(x)
-
(Af)(x)
f(x +
< \W)\\,
t)-f(x)
(Af)(x
+1)
< HeMII,
we have that Af = / ' and / € Ci(R). If / 6 Ci(R), then
ftzML - /') (x) = ^(/'(^ _ st) _ /' W)d . tt? o uniformly in z, which implies that / G D(A). Thus Af=f
for
/ G D(A) = Ci(R).
If / G D(A) and u(t) = Q(t)f, then u(t) G D(A) and u(t + h)-
u(t)
+ Au{t) = sup
f(x-h)-f(x)
+ /'(*)
h
hence w' + Aw = 0
on
[0,oo),
where u' denotes the derivative of u in the Banach space X. Note also that if v(x, t) — (u{t))(x) = f(x-t), then vt(x,t)+vx{xit)=0
for
i>0,i6l,
where t>t,t>a; denote the classical partial derivatives. EXAMPLE 4.1.4 Suppose X is a complex Banach space and B G *£(X).
" (-0*B
0(o = fc=0 £nb! It can be easily verified that
"
for CeC
-
Define
4.1.
INTRODUCTION
169
(a) 0(C) € © ( X ) , ||0(C)|| < exp(|C|||B||) for C € C (b) Q(t)Q(s) = Q(t + s) for
t,seC
(c) HC-Hl " 0 ( 0 ) " B | | < |CIHB|| 2 exp(|C|||B||) for C E C,C 7^ 0. This implies that if x G X and u{Q = Q(Qx for ( e C, then u is analytic and u' + Bu = 0. In particular, {Q(t)}t>o is a strongly continuous semigroup of operators on X and its generator is -B. EXAMPLE 4.1.5 Suppose X - LP(Rn) with p G [l,oo), n > 1. For t > 0 define (Q(t)f)(x)
= (47r*)- n/2 /
e-\x-yf^f(y)dy
for
/ G X, x G Mn
and let Q(0) be the identity map on X. In Example 3.2.13 it is shown that ||Q(*)|| < 1 for t > 0 and that (c) of Definition 4.1.1 holds. The Fubini Theorem implies that Q(t+s) = Q(t)Q(s) for t, s > 0. Therefore {Q(t)}t>o is a strongly continuous semigroup of operators on X. EXAMPLE 4.1.6 Let % be a Hilbert space, let { —00. For / G H, t > 0, define 00
(4.1)
k=l
Lemma 2.1.4 and Theorem 2.1.6 imply that {Q(t)}t>o is a strongly continuous semi group of operators on 7i and that its generator is given by 00
00
4 / = £A*(/,1P*)Wk for feV(A)
= {f€U\J2\*k(f>Vk)\2
k=l
<<*>}■ (4-2)
fc=l
One can easily verify that if / G V(A), then u(t) = Q(t)f and u' + Au = 0 for t > 0.
G V(A), Au{t) =
Q(t)Af
Observe that the point spectrum of A is equal to {Ai, A2,...} and that cr(A) = crp(A) (Exercise 2). Note that in order for (4.1) to make sense for every / G % and t > 0, we need to have that the real part of the spectrum of A is bounded from below. Theorem 2.6.8 implies that symmetric operators with compact resolvent can be represented by (4.2). A could be —A in a bounded domain with various boundary conditions, see (3.60), for a proper formulation of A. If H = L 2 (0,1), A} = -f" for / = T)(A) = W2(0,1) D WQ{0, 1) we obtain the one dimensional version studied in Examples 2.8.1, 2.8.3 and 2.8.8. In this case, the eigenvectors of A are
= v(l,t)
for = 0
t > 0, x G [0,1] for
lim / \v(x,t)-g(x)\2dx t->° Jo
£>0 = 0.
CHAPTER 4. SEMIGROUPS
170
OF LINEAR
OPERATORS
In this example, Q(t)g G n^=1V{An) C C°°(0,1) for every * > 0 and every g G L 2 (0,1) and hence Q(t) is a smoothing operator in this case. Let n = L 2 (0,1) and Af = f with domain V(A) = {/ G AC[0,1] | / ' 6 L 2 (0,1), /(0) - / ( l ) } . This operator was studied in Examples 1.6.4 and 2.6.9. In particular, the eigenvectors of A are ipn(x) = e2tn7TX, the corresponding eigenvalues are 2zmr, n — 0 , ± 1 , . . . and (4.2) holds. It is easy to see that, in this case, (Q(t)f)(x) = fp(x — t) where fp is the periodic extension of / with period 1. There is no smoothing here. EXAMPLE 4.1.7 Let % be a Hilbert space, let {2> • • •} be a complete orthonormal set in % and let Ai, A2,... be complex numbers such that, for some a G (0,00), we have that (Im A*;)2 < 4a(a + Re A*) for all k > 1. This condition implies that we can choose At* G C so that nl = Xk and |Im/Xfc| < a1/2. Define A as in (4.2). Define a Hilbert space V with an inner product [•, ■] by 00
V = {/Gft|£|/i f c (/,^)| 2
[/,p] = ^ ( i + l^l 2 )(/,^)(^,p) for
f,gev.
k=i
Define X = V xTi.
X is a Hilbert space with an inner product
({f,9h{uM)
= lfM + (9,v)
for
{/,
For t G [0,oo), define Q(t) G
= {u(t)Mt)}
G X , then where
00
u
(t) = $^((/>Wfe)cos(/i*t) +
(g,ipk)sm{iikt)/iik)
k=i 00
v
(f) = $ ^ ( (0> ¥>*) cos(nkt) - (/,
sin(iikt)nk)(pk
k=i
(set sin(fikt)/fik = t it fik = 0). {Q(£)}t>o is a strongly continuous semigroup of operators on X. Its generator, denoted by —5, is given by {/,
iff / G D(A),
If {/,(t)} = Q(t){f,9h
S{/,
then
u' = v ,
(4.3)
Backtracking, we see that the expressions for u and v make, in general, sense only when the sequence |Im/xi|, |Im/i2|, •. • is bounded, which implies that the spectrum of A has to lie within a parabola (Im A)2 < 4a(a + Re A) for some a G (0,00).
4.2. BOCHNER
171
INTEGRAL
Concretely, if U = L2(0,1), Af = /"" for / e V{A) = W4(0,1) n W02(0,1) and (pk, AA; are the eigenvectors and the eigenvalues of A, then system (4.3) is an abstract formulation of the PDE: utt(x,t) + uxxxx(x,t)
= 0 for
i>0,i6[0,l]
u(0,t) = ux{Q,t) = u(l,t) = u x (l,t) = 0 for t > 0 u(x,0) = /(z), ut(a;,0) = g(x)
for x E [0,1].
In this case, it is not obvious what (reasonable) conditions on / and g in (4.3) would guarantee that the PDE is satisfied in the classical sense.
4.2
Bochner Integral
A Banach space setting of evolution equations requires taking the derivative in the Banach space. Hence, integration of Banach space valued functions is an important tool in this setting. We shall define the Bochner integral of such functions and derive its basic properties. In the following, a subset of Rn is said to be measurable iff it is Lebesgue measur able. /i will denote the Lebesgue measure on E n . The functions will be defined on a nonempty measurable set S C Mn, with ranges in a Banach space X. x : S —>• X is called weakly measurable if 5 -> £(x(s)) is a Lebesgue measurable function for each I £ X*. x : S —> X is called almost separably-valued if there exists {yi,2/2, • • •} C X such that infj \\x(s) — yi\\ = 0 for almost all s € S. When x is almost separably-valued, it is easy to see that {yi,2/25 • • •} can be chosen to belong to the range of x. x : S —► X is called strongly measurable if it is weakly measurable and almost separably-valued. In the literature, a different definition is usually given which is then followed by the Pettis Theorem that proves their equivalence. Our approach is a bit simpler. Theorem 4.2.1 If x : S -± X is continuous at almost every point in S, then x is strongly measurable. PROOF X is obviously weakly measurable. Let C be a measurable subset of S such that fi(C) = 0 and x is continuous at each point of S\C. If S\C is empty, then we are done. Otherwise, let {ti, *2? • • •} be a dense subset of S\C and observe that, by continuity, infj \\x(s) — x(ti)\\ = 0 for each s in S\C. □ Theorem 4.2.2 Ifx:S—>Xis
strongly measurable, then s —> \\x(s)\\ is measurable.
PROOF Choose {yi, y2,...} C X and a measurable C C S such that /i(C) = 0 and infi \\x(s) — yi\\ = 0 for all s E S\C. It is enough to prove that the set V = {t G S\C | ||a;(t)|| < a} is measurable for each a G (0, oo).
172
CHAPTER 4. SEMIGROUPS
OF LINEAR
OPERATORS
Corollary 1.5.8 implies that we can choose ^1,^2, • • • in X* so that
II4II < l , iiitH) = Wvil Let Vi = {t G S\C | \£i(x(t))\ < a}. Note that V{ are measurable sets and that V C n^Vi. If t G n™iVi and e > 0, then ||x(£) - J/J-|| < e/2 for some j . Hence, \\x(t)\\
< e/2 + ||y i || = e/2 + € i (y i ) < c/2 + ^ - ( ^ - x(t))| + |^-(a;(t))| < £ + a;
therefore t £V and V = n g ^ is measurable.
□
x : 5 —► X is said to be Bochner integrable if x is strongly measurable and the function s -» ||a;(s)|| is Lebesgue integrable. The set of all such functions £ is a vector space and will be denoted by L(S,X). The following Theorem 4.2.3 enables us to define the Bochner integral fsx of x G L(S,X) to be y G X which satisfies (4.4). Theorem 4.2.3 If x G L(S,X),
then there exists a unique y G X such that
£(y) = f t(x(s)) ds
for all £ G X*.
(4.4)
Js Moreover, \\y\\ < Js \\x(s)\\ds. PROOF Assume first that there exists y G X such that (4.4) holds. Let I G X* be a normalized tangent functional to y, see Corollary 1.5.8, and note = i(y)=
fi(x{s))ds< Js
[ \\y\\\\x(s)\\ds. Js
This shows the 'moreover' part as well as the uniqueness of y. Choose {yi,j/2i • ■ ■} C X and a measurable C C S such that /x(C) = 0 and infi \\x(s) - y{\\ = 0 for all s G S\C. For *,m > 1 define A i m = {t € 5 \ C | ||x(t) - K|| < 1/m and ||x(t) - y*|| - ||x(t)|| < 0}. Theorem 4.2.2 implies that the sets Aim are measurable. Define B\m
= Aim, Bim = Aim\ Up 1 ! Ajm oo
for i > 2
4.2. BOCHNER
INTEGRAL
173
and note that xm is strongly measurable and ||#(£) — x m (£)|| < min{l/ra, ||z(£)||} for all m > 1, t G S\C. Since ||s m || = Y£i XBim\\yi\\ < 2\\x\\, we have that xm is Bochner integrable and that X ^ i M A m ) Ill/ill < 2 fs \\x\\, which enables us to define zm = Y^tLi V>(Bim)yi that satisfy OO
-
*(*m) = Y\l*(Bim)l(yi)
= / £(xm(s))ds
for all
leX*,m>l.
Since lim^j^oo fs \\xm — x\\ = 0 by the DCT, the 'moreover' part implies \\Zm ~ Zk\\ < l \\xm - Xk\\ < / \\xm -A\+ JS JS
/ ||z - Xk\\ Js
m
^ 4 ° ° 0,
which implies that zm converge to some y. Therefore the DCT implies : lim £(zm) = lim / £{xm(s))ds = / £(x(s))ds m—¥oo
m—toojc
for all
t G X*.
Jc
D
Suppose 2/1,..., yk G X and let V i , . . . , Vk be disjoint measurable subsets of S. Define x : S -> X by c
W = £xv«(*)Vi
for
teS
-
i=\
Such functions x are called simple functions. Note that the range, ft(x), is contained in {0, y i , . . . , yk}. The proof of the above Theorem 4.2.3 also yields: Corollary 4.2.4 For each x G L(S,X) there exist simple functions uk G k>l, such that R{uk) C ft(z) U {0}, \\uk{t)\\ < 2\\x(t)\\ fork>l,t€ S, lim uk(t) — x(t)
L(S,X),
for almost all t G S
k—►oo
and linifc-Kx) fs uk =
fsx.
PROOF Let x m , Bim be as in the proof of the above Theorem, with yi G R(x). For j > 1 define Vj = ^2i=l Xsim2/i, where m is such that fs \\x — xm\\ < 1/j and I is such that Y^Li+i ^(Bim)Ill/ill < Vj- N o t e t n a t Is Wx ~ vjW < 2 /j5 hence a subsequence {^jfc} satisfies lim^oo \\x(t) - vjk(t)\\ = 0 for almost all t G S. Let uk = vjk. □ This implies
CHAPTER 4. SEMIGROUPS
174
OF LINEAR
OPERATORS
Corollary 4.2.5 If x € L(S,X) and x(s) E M for all s E S, where M is a closed real subspace of X, then fsx £ M. The above Corollary and Theorem 1.7.4 imply Corollary 4.2.6 IfY is a Banach space and Q E L(S,*B(X,Y)) is such that Q(t) is a compact linear operator for each t E S, then fsQ is a compact linear operator. The Dominated Convergence Theorem generalizes to the Bochner integral: x : S -> X, / E L x (5) are such that
Theorem 4.2.7 7 / { x i , x 2 , . . . } C L(S,X),
(i) lim^oo H^tW — x(s)\\ = 0 for almost all s E S and (ii) for each i > 1 we have that \\xi(s)\\ < f(s) for almost all s E S, then x E L(Sy X) and Js x = lim^oo j s X{. PROOF It is easy to see that x is strongly measurable; hence x E L(S,X) (ii). The Lebesgue DCT and Theorem 4.2.3 imply \\ x\\Js
Theorem 4.2.8 If x E L(S,X),
Js
by
Xi\\ < / llz-ZiH -)• 0 a s % —> c o . Js
Y is a Banach space and T E *B(X,Y),
Tx E L(S, Y)
and
then
T [ x = [ Tx.
Js
Js
PROOF TX is almost separably-valued and if £ E Y*, then £(Tx) = (T*£)(x) is Lebesgue measurable; hence Tx E L(S,Y) and js£{Tx)
Corollary 4.2.9 If X,Y
= jsiTH){x)
= l{rjsv)
are Banach spaces, Q E L{S,*B{X,Y))
Qc E L(S, Y) PROOF
= (T*l) ^ x )
and
( f Qjc=
•
and cEX,
then
f Qc.
Let TA = Ac for A e *B(X,Y) and note that T E
*B{*B(X,Y),Y). D
4.2. BOCHNER
INTEGRAL
175
T h e o r e m 4.2.10 Suppose that Y is a Banach space and A : T)(A) C X ->• Y is a closed linear operator. If x : S ->• T>(A), x G L(S,X) and Ax G L(S,Y), then x e T)(A) Jo
and
A
x= JS
Ax. J S
PROOF Let ||(u,v)|| = ||u|| + ||v|| for (ti,v) G Z = X x Y. If £ e Z*, then £(u,v) = £\u + ^ for some £\ G X* and £2 £ ^*- Hence, the function s —► f(s) = (x(s),Ax(s)) is Bochner integrable in Z and / if = j hx + A £ 2 4z = £x j x + £2 f Ax = £ ( f x, f Ax) . Thus fsf = (fsx,fsAx). Let M = {(u,Au) | u G £>(A)}. Since A is closed, M is a closed subspace of Z. Corollary 4.2.5 implies that fs f G M. D Theorem 4.2.11 Suppose that (1) —00 < a < 6 < 00, a; : [a, b] —> X is continuous (2) x is differentiate
at each point of (a, b)
(3) there exists f G L x (a,6) such that \\x'(t)\\ < f(t) for all t G (a, 6). Then x' G L((a,6),X) and v{b) = x(a) + / x'(t)dt. Ja PROOF
Let x(t) = 0 for t G R\[a, 6] and define x n (t) = n(x(t + 1/n) - rc(*))
for
teR,
n> 1.
Note that x n are strongly measurable and converge to x' as n —>• 00 at each point of (a,b). Hence x' G L((a,6),X). Choose any ^ e l * and let g(t) = £(x(t)) for t G [a, 6]. Since g' = l{x% the classical result for complex valued functions implies / Ja
£(x'(t))dt = [ g'(t)dt = g(b) - S (a) = l(x(b) Ja
x(a)).
CHAPTER 4. SEMIGROUPS
176
4.3
OF LINEAR
OPERATORS
Basic Properties of Semigroups
Theorem 4.3.1 Suppose {Q(t)}t>o is a strongly continuous semigroup on a Banach space X and let —A be the generator of the semigroup. Then (1) There exist M G [0,oo), a G R such that \\Q(t)\\ < M e ' a t for all t > 0. (2) t —> Q(t)x is a continuous mapping of [0, oo) into X for every x G X. (3) IfxeX,t>0,
then /0* Q(s)x ds G T)(A) and x - Q(t)x = A /0* Q(s)x ds.
(4) If x G T)(A), u(t) = Q(t)x for t>0,
then for each t > 0 we have that
du du — (t) exists, u(t) G D(A), —(t) = -Au{t) at dt (5) {A - \)-lQ{t)
=
-Q(t)Ax.
= Q(t)(A - A)" 1 for every A G p(A), t > 0.
(6) A is a closed linear operator. (7) n^DiA*1)
is dense in X.
(8) If T G (0, oo], u : [0, r) -> X is continuous and such that du du — (t) exists, u(t) G T){A) and —(t) + Au(t) = 0 forte dt dt
(0,r),
then u(t) = Q(t)u{0) for all t G [0,T). (9) If {T(t)}t>o is a strongly continuous semigroup on X and the generator of this semigroup is equals —A, then T(t) = Q(t) for all t > 0. (10) If X is any scalar, then {extQ(t)}t>o is a strongly continuous semigroup on X and the generator of this semigroup is X — A. PROOF Let us show first that sup 0 0. If there would be no such e, then one could choose tn G (0,1/n) such that ||<2(£n)|| > n; the uniform boundedness principle (Theorem 1.4.5) would then imply that sup n ||<2(£n)#|| = oo for some x in X. However, this is not possible because, see Definition 4.1.1, limn-^oo Q(tn)x = x for every x G X. Therefore there exist M G (1, oo) and e G (0, oo) such that ||Q(t)|| < M for all t G [0, e]. Let M = e~a£. If n > 0, ne < t < (n + l)e, then ||Q(t)|| = \\Q(t - ne)Q(e)n\\ < MMn < Me~at. To show (2), it is enough to observe that for each x e l , \\Q(t + h)x - Q(t)x\\ < \\Q(t)\\\\Q(h)x - x\\
if
t>0,h>0
4.3. BASIC PROPERTIES
OF
SEMIGROUPS
177
\\Q(t - h)x - Q(t)x\\ < Me-*1-"* \\Q(h)x - x\\
if
0
(3) is obvious if t = 0. Assume t > 0 and h > 0. Then ~{1-Q(h))f
Q{s)xds
=
i / " Q(s)xds-^f
=
— / <2(s)xcte-- / n Jo ri Jh
I
= x-
Q(s + h)xds
rt
l
rt+h
Q(s)xds
rt+h
I rh Y Q(t)x + - / (Q(*)a; - x)ds - -
{Q(s)x -
Q(t)x)ds
and this implies (3). To show (4), choose h > 0 and note that for all t > 0,
r(i-QW)Q(*)^ = Q(*)r( a? -Q(^)-^QW^ as h->o. h h This implies that u(t) 6!D(A), -Ati(t) = Q(t)Ac and .D+u = - A u , where D+ denotes the right derivative. If 0 < h < £, then -\{Q{t h
- h)x - Q{t)x) =
-Au{t)
+ Q(t - h)(Ax - -(x - Q(h)x)) + Q(t)Ax - Q(t - h)Ax h and (4) follows. To prove (5), choose y G X and let x = (A — X)~1y. (A - X)Q(t)x = Q(t)(A - X)x = Q(t)y. Hence, Q(t)x = {A-
(4) implies that \)-lQ(t)y.
To prove (6), let xn G £)(A), n > 1, be such that for some x,y G X we have that lim^oo xn = x, limn_yoo Axn = y. (4) and Theorem 4.2.11 imply that xn-Q(t)xn=
Jo
Q(s)Axnds
for
£>0,n>l.
Theorem 4.2.7 enables us take the limit n —► oo. Hence, 1 -(x-Q(t)x) t
1 /** = Q(s)yds-^y t j0
as t -► 0;
thus x G £>(A), Ax = y and (6) follows. Let us prove (7) now. Choose x e X, tp e C£°(0, oo) and define poo
y(x,(p) = / Jo
(p(s)Q(s)xds.
CHAPTER 4. SEMIGROUPS
178
OF LINEAR
OPERATORS
Using Theorem 4.2.3, it is easy to see that /•oo
y(x,
rt
ipf(s)u(s)ds,
where
u(t) = / Jo
Q(s)xds.
Since Au(t) = x — Q(t)x by (3) we have that (p'Au G £((0, oo),X); since A is closed, Theorem 4.2.10 implies that y{x,ip) G T)(A) and roo
Ay{x, tp)=
tp'(s)(Q(s)x -x)ds
= y(x, tp').
Jo Since ip' is also in C£°(0,oo), we have that y(x,(p) G n ^ L ^ A " ) . Fix any x G X, e > 0. Choose t G (0, oo), fi G (0, t) so that \\Q(t)x - x\\ < e/2,
sup \\Q(s)x - Q(t)x\\ < e/2. \s-t\
Pick ip G C§°(0, oo) such that y> > 0, /x, /0°° y> = 1. Then /•OO
y(z, ip) — x = Q(t)x - x + / Jo \\y(x,tp)-x\\<\\Q{t)x-x\\
Q(t)x)ds
+ sup \\Q(s)x - Q(t)x\\ < e, |s-t|
completing (7)'s proof. To show (8), choose t G (0,r) and let v(s) = Q(t - s)u{s) for s G [0,t]. If s G (0,£], s + /i G (0,t], then v(s + h) - v(s)
=
Q(t-s-
h)(u{s + h) - u{s) - hu'(s)) +
Q(t-s-
h)u(s) - Q(t - s)u(s) - hQ(t - s)Au(s) +
hQ(t - s)Au(s) - hQ(t - s -
h)Au{s).
Therefore v'(s) = 0 for 5 G (0,*]. Since v(s) - v(0) = Q(t - s)(u(s) - w(0)) + Q(t - s)u{0) -
Q(t)u(0),
we see that v is continuous on [0,t] and therefore Theorem 4.2.11 gives that v(t) = v(0), implying assertion (8). To obtain (9), choose x G T>(A) and let u(t) = T(t)x. Prom (4) we see that u satisfies assumptions of (8), with r = oo; therefore T(t)x = Q(t)x for t > 0. This and the fact that 1>(A) is dense in X imply (9). (10) follows immediately from Definitions 4.1.1 and 4.1.2.
□
4.3. BASIC PROPERTIES
OF SEMIGROUPS
179
Theorem 4.3.2 Suppose that —A is the generator of a strongly continuous semigroup {Q{t)}t>o on a Banach space X. Let M E [0, oo), a eR be such that ||Q(t)|| < Me~at for all t > 0. Then every scalar X, with Re A < a, belongs to the resolvent set of A and, moreover, \\(A-\)-n\\<M{a-Re\)-n for n > 1, {A-X)~nx=-, (n-l)\ PROOF
—/ Jo
sn-1eXsQ(s)xds
for
n>l,xeX.
For n > 1, x E X define
Rax =
1 f°° —/ (n-l)\ JO Jo (n-1)1
sn-leXsQ(s)xds.
Clearly, Rn is a linear operator on X and, since \\Rnx\\ < -( (n-l)\
/ Jo
^ M e ^ - ^ l z l l d * = M(a - Re A)^||x||,
we have that Rn E 93(^0- We will show, by induction, that Rn = (A — X)~n. If xeT>(A),
then ^-extQ(t)x at
=
extQ{t)(X-A)x
roo
x-extQ{t)x=
/ X[o,t](s)eXsQ(s){A-X)xds for t > 0. Jo If we let t -» oo, then by the Bochner DCT, Theorem 4.2.7, we obtain that x = R\(A — X)x. Since A is closed, Theorem 4.2.10 implies that R\x E V{A), {A — X)R\x = x. This and the facts that T)(A) is dense in X, R\ is bounded and A is closed give R\y E £>(A), (A — X)R\y = y for all y E X. Therefore (A-A)"1-^!. Suppose now that Rn = (A — X)~n for some n > 1. If x E X, then ^tnextQ{t)RlX dt
= nt^e^C&t^x
-
tnextQ{t)x
tnextQ(t)Rix
= [ (nsn-1eXsQ(s)Rlx - sneXsQ{s)x)ds Jo and, by the DCT, we obtain that Rn+ix = RnR\x = (A - A) _ n _ 1 x.
□
Corollary 4.3.3 Suppose —A is the generator of a strongly continuous semigroup {Q(t)}t>o on a Banach space X. If X is in the spectrum of A and ifbeR, b > Re A, then sup t > 0 ||Q(t)a;||e w = oo for some x in X.
CHAPTER 4. SEMIGROUPS OF LINEAR
180
OPERATORS
PROOF If there would be no such x, then the uniform boundedness principle would imply that sup t > 0 ||Q(*)||e6< < oo. This and Theorem 4.3.2 would then imply that A is not in the spectrum of A. □ For example, if the spectrum of A contains any part of the half-plane Re A < 0, then {Q(t)x}t>o is unbounded for some x E X. One can show that x can be chosen so that rr E I^A), see Exercise 5. This and Examples 4.1.3 and 4.1.6 may lead one to expect that if a < inf Recr(A), then the semigroup is bounded by e~at. This expectation is generally false, as Example 4.3.4 shows. However, it will be shown in the next section (Corollary 4.5.11) that it is true in many important cases. EXAMPLE 4.3.4 Let X denote the set of all / G Cu(0, oo) such that /•OO
ex\f(x)\dx<<x>.
ll/H = sup \f(x)\+ x
Jo
X is a complex Banach space. Define Q(t) : X — ► X by (Q(t)f)(x) = f(x + t) for t, x > 0. Note that ||Q(*)/|| < ||/|| for every / G X and t > 0. If fnt(x)
= e~ni<x-^\
then
lim \\Q(t)fnt\\ = 1 = lim H/ntll, n—>oo
n—xx)
which shows that \\Q(t)\\ = 1 for t > 0. It is easy to verify (c) of Definition 4.1.1 and thus {Q{t)}t>o is a strongly continuous semigroup on X. Let —A denote the generator of the semigroup. Since for each x G (0, oo) the mapping / G X ->• f(x) G C belongs to X*, we have, by Theorem 4.3.2, that /•OO
((A - A)" 1 /)^) = / extf{x + t)dt for / G X, x > 0, A G C, Re A < 0. Jo If g = (A - A)" 1 /, then \g(x)\ < /0°° \f(x + t)\dt < /0°° | / ( t ) | * and /•OO
/.OO
rOO
/»00
/"OO
/ ex\g(x)\dx< / / ex\f(x + t)\dtdx= ex\f(x)\dx Jo Jo Jo Jo Jo
\f(x)\dx;
thus MA-Xj'ifWKWfW and hence Theorem 1.6.11 implies
for feX,
AGC, ReA<0
1} even though ||Q(*)|| = 1 for t > 0. The following Theorem 4.3.5, when combined with Theorem 4.3.2, is called the Hille-Yosida Theorem. It tells which operators are generators of Co semigroups. Its proof is rather lengthy and technical, but in essence, it is merely a justification of
\im(l+tA/n)-n
= e-tA,
4.3. BASIC PROPERTIES
OF SEMIGROUPS
181
which is obviously true when t and A are scalars. There are many other ways of constructing the semigroup from a given generator - one of which will be used in Definition 4.5.9 for a bit smaller class of operators. Theorem 4.3.5 Suppose that A is a densely defined linear operator in a Banach space X and that for some Me [0, oo), a eR, we have that (—oo,a) C p(A) and \\(A-\yn\\<M(a-X)-n
for
X G (-oo,a), n = 1,2,...
(4.5)
Then there exists a strongly continuous semigroup {Q(t)}t>o on X whose generator is equal to —A. Moreover, ||Q(*)||<Me-at
for
lim sup \\Q(t)x - (1 + (t/n)A)-nx\\ -*°°te[o,T\
t > 0,
=0
n
for
(4.6)
T G (0, oo), x G X.
(4.7)
PROOF Choose any T G (0, oo) and let K > 1 be such that K + at > 1 for a l l i G [0,T]. Note that (1 + at In)'1
(1 + at/n)~n
< KK
for
t G [0,T], n > K.
(4.8)
Since — n/t < a for t G (0,T], n > K, we may define + A)'1 = (1 + (t/n)A)-1
Rn{t) = {n/t){n/t
for
t G (0,T], n > K,
(4.9)
and let Rn{0) = 1 for n> K. Equations (4.5) and (4.8) imply that \\Rn(t)\\ < MK,
| K ( * ) I I < MKK
for
* G (0,T],n > if.
(4.10)
If x G X, y G T)(A), t G (0,T], n > K, then equations (4.9) and (4.10) imply that Rn(t)x -x = {Rn(t) - l)(x -y)(t/n)Rn{t)Ay \\Rn(t)x - x\\ < (MK + 1)||* - y\\ + tMK\\Ay\\/n.
(4.11)
Since T>(A) is assumed to be dense in X, equation (4.11) implies lim \\Rn{t)x - x\\ =0
for
n>K,xeX.
By induction, \im\\R™(t)x-x\\
=0
for
n > K, x G X, m > 1.
(4.12)
Equation (4.11) and the fact that T)(A) is dense in X also imply lim sup \\Rn(t)x-x\\ n >oo
-
iG[0,T]
=0
for
x G X.
(4.13)
CHAPTER 4. SEMIGROUPS
182
If te{0,T],t
OF LINEAR
OPERATORS
+ he (0,T], n > K, then
Rn(t + h)-
Rn(t) + (h/n)AR2n(t)
= {h/t)2(l
and (4.10) implies that R'n(t) = -(l/n)ARl(t). (R™)'(t) = - ( m / n ) A ? C
+1
M
for
- Rn(t))2Rn{t
+ h)
By induction,
t € (0,T], n>K,m>l.
(4.14)
This, (4.12) and (4.10) imply x - Rl(t)x
= [ Rl+l(s)Axds Jo
for
t 6 [0,T], n > K, x € T>(A).
(4.15)
We are now ready to prove (4.17). If y € T)(A2), t E (0,T], n>K,m>K,
then (4.14) implies
^ ( t - ^ - ^ y ^ ^ ^ ^ - ^ r H ^ C ^ - ^ ) ^
for
a€(0,t). (4.16)
This, (4.12) and (4.10) imply that RZ(t)v-K{t)y=
K+l(t-s)RT\s){--t-^)A2yds. m n
f Jo
Hence, if x £ X, then (4.10) implies
iQ(t)x - Rnn(t)x = ( i C « - i £ « ) ( * - y) + i C ( % - J$(*)y ||iCW* ~ K(*)*ll < 2MKK\\x - y\\ + T2M4K2K+2(l/n
+ l/m)||A2
and, since (4.12) implies that T)(A2) is dense in X, we have that lim sup \\IQ(t)x - Rn(t)x\\ = 0 for ' - °°te[o,T]
mn >
x £ X.
(4.17)
Since T 6 (0, oo) is arbitrary, (4.17) enables us to define Q(t) by Q(t)x = lim R„(t)x
for
t > 0, x 6 X.
(4.18)
n—►oo
This and (4.17) imply (4.7). Clearly, Q(0) = 1, and, by (4.10), we have \\Q(t)\\ < Me~at
for
t > 0.
(4.19)
Since #£(•)# : [0,T] -» X is continuous for each x E X, n > K, (4.17) also implies that Q(-)x : [0, oo) ^ X is continuous for every x e X. This and (4.19) imply \\x\\ ^ \\Q(t)x\\ < Me~at\\x\\ ^ M\\x\\ as t -» 0;
4.3. BASIC PROPERTIES
OF SEMIGROUPS
183
hence ||a;|| < M\\x\\ for each x e X. Thus, if X ^ {0}, then M > 1 and (4.6) holds; if X = {0}, then (4.6) holds automatically. (4.16) together with (4.12) and (4.10) also imply that for y e T>(A2), t2 M4
n>K,
K2K+2
\\RZ(t - s)RZ(s)y - Rnn{t)y\\ <
\\A2y\\ when 0 < s < t < T. n Therefore, Q(t - s)Q(s)y = Q{t)y for 0 < s < t, y e T)(A2). Since T)(A2) is dense in X and Q(t) are bounded, we have that Q(t)Q(s) = Q(t + s) for 5 > 0, t > 0. Therefore {Q(t)}t>o is a strongly continuous semigroup on X. Letting —B denotes its generator, (4.10), (4.13) and (4.15) imply x-Q(t)x=
I Q(s)Axds
for
Jo
t e [0,oo), x G T>{A).
This and strong continuity of Q imply that B is an extension of A. (4.5) and Theorem 4.3.2 imply a — 1 G p(A) D p(B) and hence A — B by Lemma 1.6.14.
□ The implicit Euler method for approximating the solution of u' 4- Au = 0,
u(0) = UQ
is u(t + h)-
-^
u(t)
A
.
,x
+ Au(t + h)
^
1
„
« 0,
h hence u(t + h)
«
(1 + /iA) _1 w(t)
ix(nh)
«
(l + /iA)~nw0
ii(t) « (1 + {t/n)A)-nu0. Note that Theorem 4.3.5 implies convergence of the approximations whenever —A is the generator of a strongly continuous semigroup. Theorem 4.3.6 Suppose that —A is the generator of a strongly continuous semigroup {Q(t)}t>o on a reflexive Banach space X. Then {Q(t)*}t>o is a strongly continuous semigroup on X* whose generator is —A*. PROOF Since A is closed and densely defined, so is A* (Theorems 4.3.1, 1.5.15). Let M e [0,oo), a € R be such that ||Q(*)|| < Me~at for t > 0. Then, by Theorem 4.3.2, \\{A-X)-n\\<M{a-X)-n
for
n > 1, A < a.
184
CHAPTER 4. SEMIGROUPS
OF LINEAR
OPERATORS
Hence Theorems 1.5.14 and 1.6.13 imply \\(A* - X)-n\\ <M(a-X)-n
for
n > 1, A < a
and therefore, by Theorem 4.3.5, —A* is a generator of a strongly continuous semigroup {R(t)}t>o on X*. Theorems 4.3.5 and 1.6.13 also imply {R{t)£)(x)
=
lim((l + (Vn)A*)- n Q(x) n—>oo
= =
lim £({1 + (t/n)A)-nx) t{Q(t)x) = (Q(t)*l)(x)
for all x e X, l e X*, t > 0 and therefore R(t) = Q(t)* for t > 0.
a
This, the definition of the Hilbert space adjoint and Theorem 2.2.5 imply Corollary 4.3.7 Suppose —A is the generator of a strongly continuous semigroup {Q(t)}t>o on a Hilbert space H. Then {Q{t)*}t>o is a strongly continuous semigroup on H whose generator is —A*. A strongly continuous semigroup {Q{t)}t>o is said to be a contraction semi group if ||Q(£)|| < 1 for t > 0. See Examples 4.1.3, 4.1.5 and 4.3.4 for some contrac tion semigroups, and Exercise 19 of Chapter 1 for analysis of the generator of the contraction semigroup given in Example 4.1.3. Theorem 4.3.8 Suppose that A is a linear operator in a Banach space X. Then —A is the generator of a contraction semigroup if and only if A is a densely defined accretive linear operator and Jl(A + A) = X for some A > 0. PROOF If A is a densely defined accretive linear operator and Jl(A + A) = X for some A > 0, then Corollary 1.6.17 implies that the assumptions of Theo rem 4.3.5 are satisfied with M = 1, a = 0. Suppose that —A is the generator of a contraction semigroup {Q(£)}t>o a n d that x e T>(A). By Corollary 1.5.8 there exists I G X* such that £(x) = \\£\\2 = \\x\\2 - pick any such L Define f(t) = Re£(Q(t)x) and observe that /(0) = ||z|| 2 , f(t) < ||x|| 2 , /'(*) = -Re£(Q(t)Ax) Therefore, -Ret(Ax)
for
t > 0.
= /'(0) < 0. Theorem 4.3.2 implies that A -I- 1 is onto.
□
4.3.
BASIC PROPERTIES
OF
SEMIGROUPS
185
EXAMPLE 4.3.9 Let X = {f G C[0,1] | /(O) = / ( l ) = 0} and let || • || be the sup norm on X. Assume a G C[0,1] and a(x) G (0, oo) for x G [0,1]. Define Tf=-af"
for
feT>(T)
{feC2[0,l]\fJ"eX}.
=
In Example 1.6.18 it is shown that T is an accretive linear operator and that Jl(T + A) = X for some A > 0. It is easy to show, as in Example 1.3.4, that T is densely defined. Hence, Theorem 4.3.8 implies that — T is the generator of a contraction semigroup { Q W } t > o o n X . Thus, if f eV{T) and u(x, t) = (Q(t)f){x), then (4) of Theorem 4.3.1 implies the existence of a classical solution of the parabolic equation ut(x,t)
= a(x)uxx(x,t)
for
u(Q,t)=u(l,t) lim
= Q
t > 0, 0 < x < 1 for
t>0
sup \u(x,t) — f(x)\ = 0
| u ( M ) | < ll/ll
for
*>0, 0 < z < l .
(8) of Theorem 4.3.1 gives uniqueness for the abstract equation. Uniqueness in a more classical sense can be shown by using a maximum principle (see Theorem 6.7.2 and Corollary 6.7.3). EXAMPLE 4.3.10 We shall now show that A is the generator of a contraction semigroup on X = Wo°'p(]Rn), 1 < p < oo. Note that when p < oo, Theorem 3.1.7 implies that X = Lp(Mn) and, when p = oo, Theorem 3.1.4 implies that X = Ct0. Suppose that / i , / b , . . ■ belong to the Schwartz space S and are such that A / n —> g and fn —> 0 as n ->• oo. Then [
g4>+- [
Afn= [
/nA0->O
for
<>/ G C 0 °°(E n )
and hence Theorem 3.4.1 implies that g = 0. Therefore A, as an operator in X with domain D(A) = 5 , is closable; let A denote its closure. Let us show now that cr(A) = (—oo,0] and that {\-A)-1f
= Gx*f
for
/ G l , AGC\(-oo,0]
(4.20)
where G\ is given by (3.6). This, (3.8) and Lemma_3.1.5 imply that | | ( A - A ) " 1 ! ! < 1/A for A > 0 and hence Theorem 4.3.5 implies that A is the generator of a contraction semigroup. Suppose A G C \ ( - o o , 0 ] and / G X.
There exist fn G S such that / „ - > > / . Let
un = G\* fn £ S. Lemma 3.1.5 and (3.7) imply that un ->• G\ * / and, since it is shown in Example 3.2.11 that Aun = \un - fn ->• AGA * / - / , we have that (?A*/eD(A),
(A-A)GA*/ = /
for
/ G X, A G C \ ( - o o , 0 ] .
CHAPTER
186
un
4. SEMIGROUPS
OF LINEAR
OPERATORS
n n x n x — If u G 2)(A), then there exist u G S such that u —> u and Au —» Au. As above, = G\ * (Xun — Aun) —>• G\ * (Au — Au) which implies that
u = Gx * (Au - AM)
for
u e D(A), A € C \ ( - o o , 0].
This completes the proof of (4.20) and also shows that 0 and let f(x) = e^'*-* 2 *'* for \t\2f{x)
x e
£ n . Note that f
eS,
+ (A/)(x) = (-4te 2 £ • a; + 4e 4 x • a; - 2n£ 2 )/(rc),
0<||(|C| 2 + A ) / | | p < c ( | ^ + ^ ) | | / | | p where c depends only on n and p. This shows that (A + | £ | 2 ) - 1 cannot be bounded and hence (-oo,0] C
$(u,v)=
I
JRn
Vu-Vv
for
u^eWoin
then A = —A when p — 2. T h e o r e m 4.3.8 in a Hilbert space setting is as follows: C o r o l l a r y 4 . 3 . 1 1 If A is a linear operator in a Hilbert space, then —A is the ator of a contraction semigroup iff A is m-accretive.
gener
In applications, one often encounters the following p e r t u r b a t i o n problem. You know t h a t A is t h e generator of a strongly continuous semigroup a n d t h a t t h e per t u r b a t i o n B does not effect t h e 'principle' p a r t of A + B. Is A + B t h e generator of a strongly continuous semigroup? Corollary 4.3.11 and T h e o r e m 4.3.12 provide a partial answer. For other partial answers, see Corollary 5.1.3 and T h e o r e m 4.5.7. T h e o r e m 4 . 3 . 1 2 If A is an m-accretive operator and B is an accretive operator a Hilbert space with T)(A) C ^{B), and if there exist a E [0,1) ; b < oo such that \\Bu\\ < a\\Au\\ + b\\u\\ then A + B is PROOF
for all
u E T>(A),
m-accretive. A + B is obviously accretive. If A > 0, t h e n
||(A + A ) ^ | | 2 - | | ^ | | 2 + 2 A R e ( ^ , ^ + A 2 | | u | | 2 > \\Au\\2
for
ueT>(A);
in
4.4.
EXAMPLE:
WAVE
hence \\A(A + A)
EQUATION
187
1
\\ < 1, which implies
\\B(A + A)" 1 !! < a\\A(A + A)" 1 !! + 6||(A + A)" 1 !! < a + 6/A. Therefore ||B(^4 + A ) _ 1 | | < 1 for A large enough and hence A + B is m-accretive because {A + B + A ) " 1 = (A + A ) " ^ ! + £ ( A + A ) " 1 ) " 1 .
EXAMPLE 4.3.13 The Schrodinger equation is an evolution equation in a complex Hil bert space,
w
■»,
where if is a self-adjoint operator called a Hamiltonian. ±iH are obviously accretive and, by Theorem 2.6.2, m-accretive. Thus, ±iH generate contraction semigroups. A Hamiltonian for a particle in a potential field V is defined in L 2 (E 3 ) by Hxl) = -~Kil) + Vil)
for
iPeT>(A).
In order for this H to be well defined and self-adjoint, we need to impose some restric tions on V. We shall now show that the following restrictions are sufficient: assume that V is a real valued function and that V = Vi+V2 for some Vi € L 2 (E 3 ) and V2 € L°°(E 3 ). Thus, V could, for example, be the Coulomb potential — e2(x2 + y2 4- z2)-1/2 and in this case H would represent the hydrogen atom Hamiltonian. A is as defined in Exam ple 4.3.10 with p = 2. If it G 5 , then ||Viti|| 2 < ||Vi|| 2 |M|oo- Hence, (3.14) implies that ,1/4
IIVxulb^cllVillallAull^Hul 12
(3.52) and HV2MH2 < HV2||oo||^||2 imply that for any a G (0,1), we can find b < 00 such that HVulb < a||Au|| 2 + 6|H| 2 for all u e D ( A ) . Since ±zA are m-accretive and the multiplication operators ±iV, with domain D(A), are accretive, we can apply Theorem 4.3.12 to conclude that ±iH are m-accretive. Hence R(H ± i) = L 2 (E 3 ) and therefore Theorem 2.6.3 implies that H is self-adjoint.
4.4
Example: Wave Equation
We shall first examine t h e g e n e r a l i z e d w a v e e q u a t i o n (4.23) in t h e same Hilbert space setting as in t h e section on Sectorial Forms. Hence we assume t h a t 7i is a Hilbert space, [•, •] is an inner product on V C % a n d 5 is a sectorial sesquilinear form on V such t h a t hypotheses H I , H 2 , H 3 of t h e Sectorial Forms section are satisfied. Here we assume, in addition, t h a t there exists b < 00 such t h a t \3(x,y)-${y,x)\<2b\x\\\y\\
for all
x,yGV,
(4.21)
188
CHAPTER 4. SEMIGROUPS
OF LINEAR
OPERATORS
where |rr| = [x,x]1/2. Let A be the operator associated with #. The need for an additional assumption is due to the fact that the point spectrum of A has to lie within a parabola, as demonstrated in Example 4.1.7. Assumption (4.21) guarantees that (Exercise 4). Some other consequences of (4.21) are discussed in Theorem 2.8.6 and Corollary 6.1.14. Define X = V xW. X is a Hilbert space with an inner product ({re, y}, {z, w}) = [re, z] + (y, w)
for
{rr, y}, {*, w} E X.
Define S : T>(S) -> X by D(5) = {{rr,y} | x £ T>(A),y e V} and S{x,y}
= {-y,Ax]
for
{rr,y}
eV(S).
Theorem 4.4.1 S and —S are generators of strongly continuous semigroups on X. Moreover, there exist c,M€ (0,OO) which depend only on b, a,Mi,M2,M3 such that \\(S-X)-n\\<M(\X\-c)-n
for
|A| > c, A E R, n > 1.
PROOF Note first that T)(S) is dense in X by Theorem 2.8.2. Next, define a new inner product in X by ({re, y}, {z, w})new
= 5(rr, 2;) + $(z, x) - 2a(rc, 2) + 2(y, IU)
and observe that the corresponding new norm satisfies 2min{l,M 3 }||{ a ; ,y}|| 2 < ||{x,y}|| 2 e u l < 2max{l,M 2 + |o|JM?}||{a;,y}||2. If { I . J / } 6 D(5), then
Re{S{x,y},{x,y})new \Re(S{x,y},{x,y})new\
= Re(3 r (x,y) - ff(j/,x) + 2a(y,x)) < 2(6+|a|Mi)|s|||y|| < c\\{x,y}\\2new,
where c = (6 + |a|Mi)/(2M 3 1 / 2 ). Thus, if A e R and |A| > c, then \Re((S-\){x,y},{x,y})new\>(\\\-c)\\{x,y}\\2new \\(S - \){x,y}\\new
> (|A| - c ) | | { s , y } | | n e u , . 2
(4.22)
2
If {1,9} e X, A e K and |A| > c, then - A < a + M3M^ ; hence Theorem 2.8.2 enables us to define x = (A + A 2 ) - 1 ( s - A / ) ,
y=-\x-f.
Note that {x,y} € ©(5) and (5 - A){x,y} = {f,g}. IKS-ArMlne^flAI-c)-1
for
This and (4.22) imply | A | > c , A e R,
4.4.
EXAMPLE:
WAVE
EQUATION
n
which implies | | ( 5 - A ) gives
\\new
189
< (|A| —c)
n
. Hence, t h e equivalence of t h e norms
ll(S-A)-n||<M(|A|-c)-n
for
| A | > c , A G R,
n>\
a n d some constant M < oo. Therefore, T h e o r e m 4.3.5 implies t h a t b o t h S a n d —S are generators of strongly continuous semigroups.
□
T h e o r e m 4 . 4 . 2 For each x G (A) fort>0 and u"(t) = -Au(t)
for all
t > 0.
(4.23)
Let {Q(t)}t>o b e t h e strongly continuous semigroup whose generator is —S. (4) of T h e o r e m 4.3.1 implies t h a t if x G D(-A), y G V and {u(t),v(t)} = Q(t){x,y}, t h e n for a l H > 0 we have t h a t u(t) G T)(A), v(t) G V, u is differentiable in V (and hence also in %), u' = i>, v is differentiable in % and v' = —Au. Hence we have u of T h e o r e m 4.4.2. To show its uniqueness, define v = u' and apply (8) of T h e o r e m 4.3.1. EXAMPLE 4.4.3 Consider now the initial value problem for the wave equation,
x
J
i=l j=l
i=l
'
u(x,0) = f(x),
-^{x,0)=g{x)
u(x,t) = 0
for
for
x G ft
t > 0, x G a n .
We assume that ft is an arbitrary nonempty open subset of R n . a^ = aji are assumed to be real valued, bounded, measurable functions on ft such that for some 8 > 0 n
n
Y^ Y, °v (*)&& > s\t\2
f o r a11
^ e Rn, x G n .
We assume that a* G C#(ft) for z > 1 and a 0 G L°°(ft) are complex valued. Let U = £ 2 (ft), V = W£{n) and define n
n
$(w, v) = V ] Y
Up
„
p
/ aij D iU;Dj?; + V ] / aiDiW?J + / a0wv
z=l7«
fct J=t ^
for
tu,i> G V.
^
In Section 3.7, it is shown that hypotheses H I , H2, H 3 of the section on Sectorial Forms are satisfied. Since $(w,v) — $(v,w) = JQ zv, where n
Z = Y(ai
n
+ Ui)^iW
+ X ] wD&i
+ (fl0 "" S o)w,
CHAPTER
190
4. SEMIGROUPS
OF LINEAR
OPERATORS
the additional assumption (4.21) is also satisfied and hence Theorem 4.4.2 implies ex istence of the unique generalized solution u for every / G 1)(A) and g G V. The semigroup {Q{t)}t>o actually provides a generalized solution u for every / G V and g G %; a detailed characterization of the solution in this case is given below by Theorem 4.4.6. We shall also show that a different semigroup, {R(t)}t>o, can be constructed which will enable us to define a generalized solution for every / G % and g G %. See also Example 3.2.15. Let us now examine the operator A associated with #. When a^,ao G C 1 ( n ) , Theorem 3.8.2 implies that w G V(A) iff w G W<$(Q) n w?oc(fy a n d n
n
n
Aw = -^2Y^Dj(aiJDiw)
+ 5Z aiDi ™ + a0w G L2(H).
i = l j=l
i=l
It is easy to apply the induction argument to show that D ( A m ) C W2™($1) when m > 1 and aihai G C 2 m _ 1 ( ^ ) . If 2m - n / 2 > j > 0, then W£j*(n) C Cj{Q.) by Corollary 3.6.4. This enables us to prove regularity of the generalized solution as follows. Suppose that / G T)(Ak), k>2,ge V{Ak~x) and Ak-Xg G V. Let {u(t),v(t)} = Q(t){f,g}. Note that {f,g} G 2k 1 T>(S - ). Hence, Theorem 4.3.1 implies that {u,v}^k~^ = - S 2 * - 1 ^ , * ; } and since u' = u, we have that u^2k^> exists and is continuous in H, u(t) G D(Afc) for £ > 0 and w(2fc) _ (_^4)fcw i n particular, if k > 1 4- n / 4 and the coefficients are smooth enough, then u(t) G C2(Ct) for t > 0. We shall now approach t h e generalized wave equation a bit differently. T h i s ap proach is based on t h e representation of t h e sectorial form as given by T h e o r e m 2.8.12; t h a t is, $(x,y) = (BGx,Gy) for x,yGV, which will enable us to define t h e semigroup on t h e Hilbert space Y = W, x % w i t h t h e usual inner p r o d u c t ({x, y } , {z, w}) = (x, z) + (y, w) for {x, y } , {z, w} G Y". Define T{x,y}
= {-Gy,
BGx}
for
T h e o r e m 4 . 4 . 4 T and —T are generators PROOF
{z, y} G D ( T ) = V x V. of strongly continuous
semigroups
on
Y.
Note first t h a t T h e o r e m 2.8.12 implies \\G~lx\\
< Mx\G-lx\
< MiM 3 " 1 / 2 ||a;||
If re, y € V, t h e n ff(x, y) - 5(2/, x) = (Gx, (B* - B)Gy). \(Gx, (B* - B)Gy)\
for
xeU.
(4.24)
Hence (4.21) implies
< 2b\x\\\y\\ < 2 6 M 3 - 1 / 2 | | ^ | | | | y | |
and, since 31(G) = 1-L, we have t h a t ||(B*-B)Gj/||<26M3-1/2y|
for
y € V.
(4.25)
4.4. EXAMPLE:
WAVE
Note that for x,y
EQUATION
191
eV,
2Re(T{x,y},{xiy})
=
2Re((P -
l)Gx,y)
=
Re {(B - B*)Gx, y) + Re ((B 4- 5 * - 2)Gx, y)
=
R e ( ( S - 5 * ) G a ; , 2 / ) + 2aRe(G- 1 a;,y)
since £ + B* = 2 + 2aG~2 by Theorem 2.8.12. (4.24) and (4.25) imply \*e(T{x,y},{x,y})\
< bM-1/2\\x\\\\y\\ =
2c||z||||y||
>
where c = (6+
\a\M1)/(2M^/2)
2
< \Re((T-\){x,y},{x,y})\
+ HM 1 M 3 - 1 / 2 ||x|||| 2 / ||
4{x,y}\\
(|A| - c)||{:r,y}|| 2
||(T-A){o;,y}||>(|A|-c)||{rr,y}||
for
{x,y} G D(T), A G R, |A| > c. (4.26)
If A G R and x G V, then Theorems 2.3.5 and 2.8.12 imply 0 < \\Gl'2x - \\\G-l/2xf
= ((G + A 2 *?" 1 )*, ar) - 2|A|||x|| 2 .
(4.27)
For A G R define Px = BG + A 2 G _ 1 with D(P A ) = V and note that 2PX = 2G + 2A 2 G _1 + (P + P * - 2)G + (P - B*)G. B + P * = 2 + 2aG~ 2 implies p A = G + A 2 G _ 1 + aG~l + ±(P - P*)G;
(4.28)
hence (4.27), (4.24) and (4.25) imply Re (PAo;,re) > 2(|A| - c ) | | z | | 2
for
A G R, a; G V.
Using (4.28) and the fact that G is m-accretive gives that PA +
M
= (l + ((A 2 4-a)G- 1 + i ( P - P * ) G ) ( G + M)"1)(Gr + M) for /x > 0.
Hence, (4.25) and Theorems 2.3.2, 1.6.8 imply that #(P A + fj) = U for /* large enough and therefore P A — 2(| A| — c) is m-accretive and, in particular, 0 G p(P\) when A G R, |A| > c. Therefore, if {/,#} G Y", A G R, |A| > c, we can define x = Px-1(g-\G-1f),
y = -G~1(\x
+ f)
and note that {x,y} G D(T) and (T-A){rr,y} = {/,#}. This and (4.26) imply that ||(r — A)"1)] < (|A| — c ) - 1 for | A | > c , A G R and therefore Theorem 4.3.5 implies that both T and —T are generators of strongly continuous semigroups o n Y . d
CHAPTER 4. SEMIGROUPS
192
OF LINEAR
OPERATORS
Let {R(t)}t>o be the strongly continuous semigroup on Y whose generator is —T. Theorem 4.4.5 If x E V, y E U and {u(t),v(t)} = Q{t){x,y} R(t){x, G~ly} = {u(t), G^vit)} for t > 0.
for t > 0, then
PROOF (3) of Theorem 4.3.1 and Theorem 4.2.8 imply that x - u = - fQ v, y — v = AJQU. Hence v! = Gw, where w = G~lv and G~ly — w = BG J0 u. Since u is continuous in V, Theorems 2.8.12 and 4.2.10 imply G~1y—w = J 0 BGu and thus (8) of Theorem 4.3.1 implies the assertion of the Theorem. D Theorem 4.4.6 / / x E V, y € U and {u(t),w(t)} = R{t){x,G-ly) for t > 0, then thisu is the unique u E C([0, oo), V)flCfl([0, oo),H) which satisfies u(0) = x, u'(0) = y and ^ 2 ("(*),*) = ~d(u(t),z)
for
t > 0, z E V.
PROOF Theorem 4.4.5 implies that u E C([0, oo), V). (4) of Theorem 4.3.1 implies that u' = Gw, w' = —BGu and hence for every z E V, we have that (tt, z)' = (uf, z) = {Gw, z) = (w, Gz). Thus, (w, z) is actually twice differentiable and (u,z)" = (w',Gz) = ~{BGu,Gz) = -${u,z). To show the uniqueness, let w = G~lu' and note that w E C([0, oo), V), u' = Gw and that for every z E V we have (Gw,z)' — (w,Gz)r = —(BGu,Gz). Hence (w(t),Gz) - (w(Q),Gz) = - f Jo
{BGu,Gz)
and, since the range of G is %, we have that w(t) — w(0) = — fQ BGu. Therefore, w' = —BGu and hence (8) of Theorem 4.3.1 implies the uniqueness. □
4.5
Sectorial Operators and Analytic Semigroups
Theorem 2.8.2 implies that any operator A associated with a sectorial sesquilinear form satisfies the assumptions of the Hille-Yosida Theorem 4.3.5 and hence defines a semigroup that can be denoted by e~At. Since such operators A generalize elliptic operators (Section 3.7), we can say that ut + Au = 0 generalizes parabolic equations and e~Atu(0) is the solution of the initial value problem. However, more can be said in this case. In view of Figure 2.1 and The orem 1.6.16, it is easy to see that — £A is also a generator of a strongly continuous
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
193
semigroup when ( G C and |arg(£)| is small enough. Hence, it is natural to ask: Is e~A*> an analytic function of f ? Note that semigroups need not be differentiable in general (Example 4.1.3). We shall prove that the answer is yes in this case. Further more, we shall show that the analyticity also implies smoothing of solutions, which was observed in Example 4.1.6 and is a characteristic of parabolic equations. To avoid being constrained to Hilbert spaces, we shall first define a bit larger class of operators for which this analysis can be done. We shall then show some properties of these operators followed by a direct construction of e~A*> and its analysis. Complex Banach spaces will be used in this section. In the next section it is shown, Corollary 4.6.2, how to proceed when real function spaces have to be used. Let arg(O) = 0 and, if C G C\{0}, let arg(C) G [—TT,TT) be such that C = |C|e i a r g ( c ) . Definition 4.5.1 For a G E, M G (0, oo), 0 G (0,7r/2) and a complex Banach space X, define 2t(a, M, 9,X) to be the collection of all densely defined linear operators A in X which have the property that Xep(A)
\\(A-X)-l\\<M/\X-a\
and
whenever A G C and | arg(A — a)\ > 6. A is said to be a sectorial operator if A belongs to some 2l(a, M, 0, X). EXAMPLE 4.5.2 If A is the linear operator associated with a sectorial sesquilinear form on a complex Hilbert space, then Theorem 2.8.2 implies that , IK
'
f p ^
if | a r 6 ( A - a ) | > 0 + ,r/2
" ~ I U-I^I^O-.)!-*)
tf«
+ T/2
_1
where 9 = tan (M2/M 3 ). Hence, A is a sectorial operator. EXAMPLE 4.5.3 Consider - A in W°' p (R n ), 1 <_p < oo, see Example 4.3.10. Observe that (4.20), (3.7) and Lemma 3.1.5 imply that - A is a sectorial operator. EXAMPLE 4.5.4 For / G Ci, define 1(f) = lim| x |_^ 00 /(x). Define
Atf = A(f-i(f))
for feV(At)
=
{feCe\f-i(f)eV(K)},
where A is defined as in Example 4.3.10 when p = oo. Observe (Exercise 7) that A^ is densely defined, cr(A^) = (—oo,0] and (X-At)-1f
= Gx*f
for feCi,
AGC\(-oo,0]
where G\ is given by (3.6). This implies, as in Example 4.5.3, that -A* is a sectorial operator in Ci and that A* is the generator of a contraction semigroup on Ci. Lemma 4.5.5 If A is a sectorial operator and if for some A E R the line Re£ = A, ( £ C, lies in the resolvent set of A, then for some 8 > 0, the strip |A — Re£| < 8, C G C, also lies in the resolvent set of A.
CHAPTER 4. SEMIGROUPS
194
OF LINEAR
OPERATORS
PROOF Suppose A G 2l(a,M,0,X). If there would be no such S > 0, then there would exist a sequence {CAJ in o~(A) such that ReOfc —> A and, since |ImCfc| < (Re(A; — a ) t a n 0 , we see that a subsequence of {(k} should converge to some ( with Re£ = A - this is not possible since a (A) is closed. D T h e o r e m 4.5.6 Suppose that A is a sectorial operator in X and that for some a G IR, a
for all A G o~(A).
Then A G2l(a,M,0,X) for some Me PROOF
(0,oo)? 0G (0,?r/2).
Suppose A G 2l(ai, Mi, 0\, X). If a < a\, then |A — a i | > |A — a|sin#i
when
| arg(A — a)\ > 0i;
hence A G 2l(a,Mi/sin0i,0i,X). If a > ai, then, by Lemma 4.5.5, there exists b > a such that b < Re A for all A G cr(A). Define 0 by (b — a) tan0 = (b — a\) tan0i and note that S = {( G C | |C - a i | < {b- ai)/cos0i,ReC < 6} C p(A). Hence, |£ — a|||(A — C) _ 1 || is & continuous function of ( G 5 and therefore has a finite maximum c\ in S. Thus \\{A - \)-l\\
for
A G 5.
If A 0 # and | a r g ( A - a ) | > 0, then | A - a | < | A - a i | + a - a i < 2 | A - a i | . Hence choosing M = max{ci, 2M\} will do it. □ T h e o r e m 4.5.7 Suppose A G 2l(a,M,0,X), B : T>(B) D V(A) -> X is linear and \\Bx\\ < e\\Ax\\ + c\\x\\ for
x G T>(A),
w/iere 0 < e < l / ( l + M) ana7 c < oo. T/ien, ^4 -f B is a sectorial operator in X. PROOF Choose r so that /i = e(l + Af) + (e|a| 4- c)M/r < 1. If |A - a| > r and | arg(A — a)\ > 0, then
\\B(A-X)~l\\
(A + B-X)-1
< sWAiA-Xr'W+cUA-Xr1]] < e (l + ||A(^-A)- 1 ||) + c||(^-A)- 1 || < e{l+M) + (e\a\+c)M/\X-a\
11(^1 + 1?-A)-11| < —^L
-;
(l-fj.)\X-a\
hence A + B e?l{a-r/sin0,M/(l
~n),0,X).
a
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
EXAMPLE 4.5.8 Let A be defined as in Example 4.3.10 in X = W^p(Rn), Note that (3.16) implies that for some c < oo, ||D 4 u|| p < c | | £ t < a | | u | | J / a <e\\Au\\p
+ c2/{Ae)\\u\\p
195
1 < p < oo.
for fc = l , . . . , n ,
for every e > 0 and every u G <S - and hence, by Theorem 3.4.3, also for every u E T)(A). Theorem 4.5.7 implies that J
n
Tu = - A M + ^
aiDiU + aQu
for
u € T>(T) = D(A)
i=i
is a sectorial operator in X , provided that a; 6 L°°(E n ). When p = oo, we require also that di are continuous. For b G R,
for
(4.29)
£G !
Figure 4.1: Integration p a t h 7. Let <3T denote t h e closed p a t h consisting of straight lines from 7(0) to 7 ( T ) to y(—T) to 7 ( 0 ) . T h e Cauchy formula implies t h a t for any polynomial P a n d any ( G C,
-f
P(A)e -AC -dX
-{
0 P(z)e~('z
if z is outside < 7 if £ is inside <\T
If C 7^ 0 a n d 6ReC + |tC|cos(
for
t G E,
(4.30)
CHAPTER 4. SEMIGROUPS
196
OF LINEAR
OPERATORS
which implies that we can take the limit T -> oo to obtain for z G C\{6}, 1
/■«> P ( 7 ( t ) ) e - ^ K
2^ loo
T(*)-*
7w
_ J o
" I n*)^
if | arg(* - 6)| > ^ if ar
I e(* - *)l < v '
l
,
j
This motivates the following: Definition 4.5.9 For A G 2l(a,M,0,X) and C G C with |arg(C)|
define
Z" 0 0
1
iy
(7(i) A le 7(t)
" r ~
VM
(4-32)
b G (-00,a) and (p G (0,ir/2 - |arg(C)|).
(4.33)
where 7 is given by (4.29), see Figure 4.1, with
To justify the definition, observe first that |7(*) -a\>(a-b)
sirup, | arg( 7 (*) - a)| > y? > 0;
(4.34)
hence 7(2) G p{A) for t 6 E and the integrand in (4.32) is continuous on M\{0} and is therefore strongly measurable. This and (4.30) imply oo
/
||( 7 (t) - A ) - 1 e - ^ ^ V ( t ) | | d t <
jjifp-bReC, 7
LN„,
,
=
—
(4.35)
-00 - (a-6)|C|sm(^cos((p + |arg(C)|) and therefore the Bochner integral in (4.32) exists. Let us show now that the value of the integral in (4.32) does not depend on the choice of 6,ip in (4.33). Let I(b,(p) denote the integral. Choose £ G 93(X)* and note that
l(I(b,
[f(\)d\,
for | arg(A - a)\ > 0. For T > 0 define E(T) by rT
l(I(b, 00. Choose 61 G (-00, a) and <^i G (0,7r/2 — | arg(()|) and let 71, Ei denote the corresponding 7, E. Theorem 1.6.11 implies that / is analytic. Hence, the Cauchy Theorem gives that *(/(&,¥>)) - l(I(bi,
+ F(T),
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
197
where F(t) = [
f(X)d\ = f f(ay(t) + (1 - s) 7 i(t))( 7 (t) ~ 7i(t))da.
Observe that for all |t| large enough, we have that |57(*) + ( l - * ) 7 i ( * ) - a | > l and hence (4.30) implies that \f(sy(t) + (1 - «)7i(*))l < |K||Mexp(-cReC - |«CI costy + | arg(C)|)), where c = min{6,6i}, ip = max{<£,ipi}. Therefore F(t) converges to 0 as \t\ —► oo which implies £(I(bi,(fi)) = i(I(b, does not depend on the particular choice of a, M or 6 either. Hence, e~A<> depends only on £ and on the sectorial operator A. Note that the above argument could be applied to show that many other integra tion paths could be used in Definition 4.5.9 without changing e~A(*. Minimization of (4.35) with respect to b E (—oo,a) gives Theorem 4.5.10 // A E 2l(a,M,0,X), C G C\{0} and | arg(C)| < TT/2 - 0, then M-ACM <
Mecos(arg(0) inf e_aReC (pG(0,7r/2-|arg(C)|) n sinip cos(tp-\- |arg(C)|)
This and Theorem 4.5.6 imply Corollary 4.5.11 If A is a sectorial operator and a E R. is such that a < Re A for all A E o~(A), then there exist M < oo and 6 > 0 such that \\e~M\\ < Me- f l R e C
for all (eC
with | arg(C)| < £
Theorem 4.5.12 If A E 2l(a,M,0,X), tfien e~Awe~Az = e~A^w^ for all complex numbers z and w which satisfy | arg(z)| < TT/2 — 6, \ aig(w)\ < TT/2 — 6. PROOF We may assume that z ^ 0, w 7^ 0. Let a = max{| arg(^)|, | arg(u>)|}. Choose b E (—oo,a) and (p E (0,7r/2 —a) and define j(t) = b+\t\ cosy? — it sincp, /i(t) = j{t) + (a - 6)/2 for t E R. Choose £ E
/ oo
/ -oo
e(e-Aw{y{t) -
A)~1)e-'^zy'(t)dt
-oo
roo
dt / ds<(o*(a) -A)-1(7W - A r v ^ ' ^ W O O V—oo
CHAPTER 4. SEMIGROUPS
198
OF LINEAR
OPERATORS
Using the resolvent identity, (/* - A)~l - (7 - A)'1 = ( 7 - p)<jx - A)-l(j
A)'1,
-
and the Pubini Theorem gives
(2-Ki)2e(e-Awe-Az) = ~ ( r ~ * - « * ! ^ )
/ : ( /
/*(«)-7(*)
e M t )
_
A )
- i
) e
^
m
)
The Cauchy formula (4.31) implies
/
oo -co
= (2m)2£(e'A^w+z^) and, since this is true for all £ E ©(A")*, we have that e~ Aw; e- A * = e-A(™+z) by Corollary 1.5.10. □ This proves the semigroup property of e~A<*. Analyticity of e~A<> follows from Theorem 4.5.13 If A e 2l(a, M,0,X), C € C\{0} ? |arg(C)| < TT/2 - 6, then for n>0,
where 7 is as in Definition 4-5.9 of e~A^. PROOF The integral, denoted by In{Q, clearly exists. If h e C and \h\ < |C|cos(
h->0
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
199
The smoothing property of e A^ is the consequence of the following: Theorem 4.5.14 If A G 2l(a,M,0,X), C G C\{0}, |arg(C)| < TT/2 - 6, then for n>0, (1) the range of e~AC> is contained in T>(An) (2) Ane-A(
=
(3) Ane~A^x
{-±)ne-A(
= e~A^Anx
for
x G V(An).
PROOF Statements (1) and (2) are obviously true for n = 0. If (1) and (2) are true for some n, then Theorem 4.5.13 implies that for all x G X we have Ane~ACx
1 r°° = -=T / 7(t) n ( 7 (t) -
A^xe-^-y'^dt.
Since A is closed and A(^(t) - A)~lx = -x + -y(t)(j(t) - A)~lx, Theorem 4.2.10 implies that Ane~AC>x G Tf(A) and A^e-^x
= -^ /
2TTZ J_OQ
j(t)n(-x
A)-1x)e-^t^j'(t)dt.
+ 7(*)(7(*) -
The Cauchy formula (4.31) implies An+1e-A^x
= -^ / 7W n + 1 (7(*) " 2m J_00
A)-lxe-^^i{t)dt.
Therefore, (1) and (2) are true for n + 1 and hence for all n > 0. Assuming (3) for some n > 0 implies that for x G T){An), 1 f°° Ane~A<x = - ^ / (7W 2m J_00
A)-lAnxe-^h'{t)dt.
Hence, if x G D(A n + 1 ), then Theorem 4.2.10 implies (3) for n + 1. Theorem 4.5.15 Suppose A G 2l(a,M,0,X) and £ G (0,TT/2 - 0). Z>e/me ^-{CGC\{O}||arg(C)|<7r/2-0-£}. Then, (1) lim^o,ce5 | | e ~ ^ x — x\\ = 0 for each x G X (2) Umc_>0>C€5 IIf(a? ~ e ^ x ) - Ax\\ = 0 /or eac/i x G D(A)
□
CHAPTER 4. SEMIGROUPS
200
OF LINEAR
OPERATORS
At
(3) {e
}t>o is a strongly continuous semigroup whose generator is —A.
PROOF e-A<;y
Choose y G 2)(A), £ G S. If 7 is as in Definition 4.5.9 , then _ e-oCy
=
— 1
( ( 7 « ) - A)-ly
/
- ( 7 « - a)-ly)
e~^W(t)dt
r°°
Mt)-a)-1h(t)-A)-l(Ay-ay)e-^
= 2^J_
Using (4.34) and (4.30), with b = a - 1/|C| and ip -* 9, gives |, e -AC. _ c-«Cw|i < | C . Me\\Ay-ay\\
ReC
This, the bound in Theorem 4.5.10, the fact that \\e~A<x - x\\ < \\e-A
C
= Cf e-^Axdt Jo r1 = / {e~A^Ax - Ax)dt.
Jo
This and (1) imply assertion (2). Let —B be the generator of the semigroup {e~At}t>o- (2) implies that B is an extension of A. Theorems 4.5.10 and 4.3.2 imply a — 1 G p(A) C\ p(B) and hence A = B by Lemma 1.6.14. D EXAMPLE 4.5.16 Consider the initial value problem for the parabolic equation: du ~dt
S S ^ ~ \aii^dx') ~^ai^~dx- ~a°(x)u u(x,0) = /(x)
for
* > °> x e
fi
for I G H
u(x,t) = 0 for i > 0 , x e dft. Assume that Q is a nonempty open subset of Kn, that a ^ , ^ are bounded, complex valued, measurable functions on Q such that Ima^ = Im aji and that the strong ellipticity condition (3.62) holds with some S > 0. Define the sectorial form by (3.61) on
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
201
V = Wo1 (ft), H = L2(ft) and let A be the sectorial operator associated with it. The generalized solution of the parabolic equation is u(x,t) = (e~Atf)(x) for / G L2(ft). Theorem 4.5.13 implies that t ->• u(-,t) is an analytic, Hilbert-space-valued, func tion. Hence, t —> (u(-,t),v) is a complex valued analytic function for any v G L2(ft). Note that when v is a characteristic function of a compact set C C ft, then (u(-,t),v) represents the average of u(-,t) in C multiplied by the volume of C. When aij,a0 G C^ft), Theorem 3.8.2 implies that w G T)(A) iff w G W£(Q) D W2oc(Q) and n
n
n
Aw = - J Z 5Z DjfaijDi™) + ^2 ciiDiW + a0w G L2(ft). 1=1 j = l
1=1
It is easy to apply the induction argument to show that V{Am) C W2™(Q.) when m > 1 and a^-,^ G C 2 m _ 1 (n). Corollary 3.6.4 implies that ^2™(ft) C C'(ft) if 2m - n/2 > j > 0 and Theorem 4.5.14 implies that u(-,t) G n£ 1 D(A / : ) for £ > 0. Thus, if aij,ai G C°°(ft), then, for t > 0, we have that w(-,t) G C°°(ft) for any / G L2 (ft) and therefore e_A* is a smoothing operator. Corollary 4.5.11 implies that if the spectrum of A lies in the half-plane Re£ > 0, then u decays exponentially in t. On the other hand, Corollary 4.3.3 implies that if the spectrum of A contains any part of the half-plane Re C < 0, then u is unbounded in t for some / G L2(ft). This stability-instability result is true for any sectorial operator. Let B be a bounded operator on a complex Banach space X. Since B is a sectorial operator (Exercise 12), we have e~B^ defined for C in a sector. How does e~B^ relate to Q(C) which is defined in Example 4.1.4 for every ( € C? Since {Q(t)}t>o is a strongly continuous semigroup whose generator is also — B, Theorems 4.3.1 and 4.5.15 imply that Q(t) = e~m for t > 0. £(Q(-)x) and i{e~B'x) are complex-valued analytic functions in the sector and, since they are equal on (0, oo), they are equal in the whole sector. Moreover, since this is true for every x G X and £ G X*, we have that Q(C) = e~B<> for all C in the sector. This also enables us to extend the definition of e~B^ to every C € C, by e-B{ = y t ^ B
k
for
(GC
We shall often need the following bound on Ae~A<> Theorem 4.5.17 If A G 2l(a,M,0, X), C € C\{0}, |arg(C)| < TT/2 ~ 09 then ||(A-a)e-*||<
M e" a R e ^ 7rcos(0 + |arg(C)|) |CI
(4.36)
CHAPTER 4. SEMIGROUPS
202
PROOF
OF LINEAR
OPERATORS
Theorems 4.5.13 and 4.5.14, with 7 as in Definition 4.5.9, imply that
(A-a)e-A<
1
=
r°°
— y_
\\(A -a)e~A<\\ < l±l
( 7 (t)-a)( 7 (t)-A)- 1 e-rt«XyW«ft
Je~^\dt ^_
"
e-^ReC.
7r|C|cos((p + |arg(C)|)
Letting b —> a and ip —>• 6 gives the bound.
□
Corollary 4.5.18 If A is a sectorial operator and r € (0,00), then for some c < 00, \\Ae-At\\ < ct~l
for
||e-At_e-As|| < C 5 - i ( t _ 5 )
0 < t < r, /or
0<5
||yle-A* - A e - ^ H < ct- 1 ^- 1 ^ - 5) for
PROOF
(4.37)
0<s
(4.38) (4.39)
Theorems 4.5.10 and 4.5.17 imply (4.37). Theorem 4.5.14 implies
||e-At_e-A*|| < I pe->lr||dr and hence (4.37) implies (4.38). Since \\Ae~At - Ae~As\\ < f \\A2e~Ar\\dr < f
\\Ae~Ar'2\\2dr,
(4.37) implies (4.39).
D
The bound in (4.37) is a defining characteristic of analytic semigroups. This is due to the fact that if —A is the generator of a strongly continuous semigroup {Q(t)}t>o, such that £||AQ(£)|| is bounded for 0 < t < 1, then A is a sectorial operator. The proof of this fact consists of the justification of the following representation:
-AC
=
y^ (* ~ On (Ac-At/n)n. 71=0
We shall not need this result, so no more details will be presented. Theorem 4.5.19 Let {Q(C)}<es be a family of bounded operators in a complex Banach space X where S = {( G C | |arg(()| < /3, C 7^ 0} and f3 € (0,7r/2). Assume that
4.5. SECTORIAL
OPERATORS
AND ANALYTIC
SEMIGROUPS
(a) for some M < oo and a G R, we have that \\Q(()\\ < M e _ a R e C
203
for(eS
(b) ( —>• £(Q(()x) is an analytic function in S for every I G X*, x G X (c) {Q(t)}t>o is a strongly continuous semigroup whose generator is —A. Then A is a sectorial operator in X and e~A<* = Q(() for ( G S. PROOF Choose I G X*, x G X, A < a and let /(C) = £{Q{()x)ex( Since / is analytic, the Cauchy Theorem implies that
f f(t)dt+f Je
/(CK= / f{zt)zdt+ [ JT-±Tz
for 0 < e < T and z G S. e -» 0+ and obtain
Je
for ( e S.
/(c)dc
J e-^ez
(a) implies that we can take the limits T —>• oo and /•oo
/
A'OO
f{t)dt=
Jo
/
f(zt)zdt.
Jo
Theorem 4.3.2 then implies that for all I G X*, x G X, we have /•OO
*((A - A)-Xx) = z I /o
£(Q(zt)x)eXztdt
for
z G 5, A < a.
(4.40)
(a) implies that for every z G S the right-hand side of (4.40), denoted by RHS(\)Z, is an analytic function of A in the half-plane Re (A — a)z < 0. (4.40) implies that RHS(X)Z = RHS(\)i for A G (-co, a) and hence RHS(X)Z = RHS(X)\ for all A in their common domain. Thus, RHS(X)i can be extended to an analytic function of A in the region where Re (A — a)z < 0 for some z G S, i.e. | arg(A — a)\ > 7r/2 — /3 and in this region we have the bound
\RHSim
< inf I'MI'im*
zes —Re (A — a)z M\\t\\\\x\\ -|A - a\ cos(| arg(A - a)\ + 0)'
(4.41)
Let a be the smallest nonnegative number such that if | arg(A — a)\ > a, then A G p(A). Theorem 4.3.2 implies that a < ir/2. Since (A - A) - 1 is analytic, we have that t{(A-X)~lx) = RHS(X)i in the sector | arg(A-a)| > max{a,7r/2-/3} and hence (4.41) implies (see Exercise 11 of Chapter 1) " ^ - ^ ^ - l A - a l c o s d ^ A - a ) ! ^ )
^
in the sector. If a > n/2 — /3, then Theorem 1.6.11 implies that, at each point A in the sector | arg(A — a)\ > a, a disk of radius — |A — a\ cos (a + (3)/M also
CHAPTER 4. SEMIGROUPS
204
OF LINEAR
OPERATORS
lies in p(A) and hence a could not be the smallest. Thus, a < 7r/2 — /3 and (4.42) implies that for every 0 6 {TT/2 - /?, n/2) there exists Me < oo such that A E 2l(a, Me, 0, X). e~A<> is then defined and analytic in S and, since Theorems 4.5.15 and 4.3.1 imply that e~At = Q(t) for t > 0, the analyticity implies £(e-A^x)
= l{Q{Qx)
for all
( E S, £ £ X\
x 6 X.
Hence Corollary 1.5.10 implies that e~At = Q{() for ( e S.
4.6
□
Invariant Subspaces
A closed subspace Y of a Banach space X is said to be an invariant subspace of an operator A, defined in X, if Ay € Y for every y £ 1)(A) fl Y. In this case, the restriction of A to Y is defined to be an operator AiY in Y given by AiYy
= Ay
for
y € D ( i i y ) = X>(A) H Y.
Theorem 4.6.1 Lei {Q(t)}t>o be a strongly continuous semigroup on a Banach space X, let -A be its generator and let a,M eR be such that ||Q(t)|| < Me~at for t>0. IfY is an invariant subspace of (A — A) - 1 for some X < a, then Y is an invariant subspace of A and Q(t) for all t>0. Moreover, {Q(t)iY}t>o is a strongly continuous semigroup on Y whose generator is — AiYPROOF
If |/z — A| < a — A, then Theorem 4.3.2 implies that | | ( / i - A ) n ( A - A ) - n | | < M|/i - A| n (a - A)~n
for
n > 1
and, since Y is closed, (1.22) and Theorem 1.6.8 imply that Y is an invariant subspace of (1 — (/i — X)(A — A ) - 1 ) - 1 . The resolvent identity, (A -
M)-i(l
- (M - \)(A - A)" 1 ) = (A - A)~\
implies that Y is an invariant subspace of (A — fi)~l for \x E (A — a + A, a) and, by induction, for all \i < a. If t > 0, then -n/t < a for large n. Hence if y G Y, then (A + n/t)~ny E Y and, by Theorem 4.3.5, Q{t)y G Y. Thus, Y is an invariant subspace of Q(t) and {Q(t)iy}t>o is obviously a strongly continuous semigroup on Y whose generator is — AiY. □ When working with real evolution equations, one would expect that real initial conditions would produce real solutions. To ensure this one can always try to set up the problem in a real function space. However, the preparatory analysis is often
4.6. INVARIANT
SUBSPACES
205
more easily done by using a complex function space X. The real parts do not form a subspace of X\ however, a complex vector space X can always be considered to be a real vector space and thus the real parts do form a real subspace of X and hence Theorem 4.6.1 applies. For example, Corollary 4.6.2 Let A be a sectorial operator in a complex Banach space X. Con sider X to be a real Banach space. Let Y be a closed real subspace of X and suppose that Y is an invariant subspace of (A + A) - 1 for some sufficiently large A G M . Then Y is an invariant subspace of e~At for t>0. EXAMPLE 4.6.3 In Example 4.5.3 it was shown that —A is a sectorial operator in X = WQ iPJ^Ln). Let Xr be the set of real valued members of X. Prom formula (4.20) for (A — A ) - 1 we see that Xr is an invariant real subspace of (A — A ) - 1 , A > 0. Corollary 4.6.2 implies that Xr is an invariant real subspace of eAt for t > 0 and hence {(eAt)±xr}t>o is a strongly continuous semigroup on Xr whose generator is A^x,,Actually, {(eAi)4,xr}t>o is a contraction semigroup on Xr, see Example 4.3.10. The same arguments apply to A*, see Example 4.5.4, when one wants to consider it in the space consisting of only real members of Ci. A bounded operator P on a Banach space is said to be a projection if P2 = P. The following two Lemmas, Theorem 2.2.3 and Exercise 13 highlight the relationship between projections and closed subspaces. Lemma 4.6.4 When P is a projection on a Banach space X, then (a) 1 — P is also a projection on X (b) # ( P ) = N(l - P) is a closed subspace of X (c) !R(1 — P) = N(P) is a closed subspace of X (d) # ( P ) n f t ( l - P ) = {0} (e) X = # ( P ) + tt(l - P) (f) 0i(P), ft(l - P) are invariant subspaces of B if B e
206
CHAPTER 4. SEMIGROUPS
OF LINEAR
OPERATORS
Lemma 4.6.5 If P is a projection on a Banach space X and A is a linear operator in X such that (A - X)-XP = P(A - A)" 1 for some A <E p{A), then, ft(P) and #(1 - P) are invariant subspaces of A and (A - C)" 1 for every C € p(A). PROOF If x e T>(A)nR(P) and y = (A-X)x, then x = Px = P{A-X)~ly l (A - X)~ Py. Hence {A - X)x = Py and Ax = Py + Xx e tt(P).
=
If x 6e B(P), C e p{A) and y = = (A (A- - O " 1 * , then Py = P(A - Xy^A
1 - X)y = (A - A A)-) X"P(A ^ - X)y <E T>(A),
hence, {A - X)Py = P(A - X)y. Therefore, {A - ()Py = P(A - Qy = Px = x andPy = yeR(P). This gives the invariance of tt(P). (A - A ) - 1 ^ - P) = (1 - P)(A - A)" 1 implies the invariance of ft(l - P ) . □ Recall that a cycle T is a finite collection of closed paths 7 i , . . . , 7 n in C and that the integration over T is defined by
f/ f(Qd( /(CK == J2 J2 ff f(Qd(. f(Qd(. If each path 7fc lies in some open set ft, we say that T is a cycle in ft. Let K be a nonempty compact subset of an open set ft in C. It is well known that a cycle T can be chosen in Q\K so that
l
/* c?C _ / i
2TTZ y r rc(;-X~\ - A ~ \ 2
o0
if A e nr
if A 0 ft ifA0ft
'
(4.43) [ 6)
^
Any such cycle T is called a contour that surrounds K in ft. If T is another contour that surrounds K in ft, then the Cauchy Theorem states that for any function / that is analytic in Q\K, we have / //(CR (CR = = / /(C)dC/(C)dC7r Jr Jr Jr
(4.44)
A nonempty compact set a in C is called a spectral set of a linear operator A in a complex Banach space X if a C a(A) and a(A)\cr is a closed set in C. In this case define (4.45) P ^ ^ / / - A ^ where T is a contour that surrounds a in (a(A)\a)c. Since {a{A)\a)c\a = p(A), the Bochner integral in (4.45) exists and Pa G
4.6. INVARIANT
SUBSPACES
207
Theorem 4.6.6 Let a be a spectral set of a linear operator A in a complex Banach space X and let Pa be given by (4-4$) • Then (1) Pa is a projection on X (2) {A - A ) " ^ = Pa(A - A)" 1 for every A G p(A) (3) 0l(Pa) and CR(1 — Pa) are both invariant subspaces of A (4) R(Pa) C
G ®(B(P«r)) and O-(AWP(T))
=a
=
PROOF Choose 8 > 0 such that a + B(0,8) c (a(A)\a)c. that surrounds o in cr + P(0,8) and hence also in (a(A)\a)c. that surrounds a + JB(0, 8) in (a(A)\a)c. Then
Let T be a contour Let V be a contour
and the resolvent identity implies
A) 1
*-<£VLLM- - -«-
A)
-i,dCd(
'c-c
This, the Fubini Theorem and the fact that (4.43) implies
i/rrc=°for^^' h!T,
and A2 — Ai-R{x _ P
(4.45) and Theorem 4.2.10 imply that ^(Pa) C T)(A) and if x G $.{Pa), then \\A1X\\ = \\AP0x\\ = \\^- f A(C ~ A)-xxdC (A2). The same holds on Jl(Pa)', hence p(A)cp(Al)nP(A2). (4.46)
CHAPTER 4. SEMIGROUPS OF LINEAR
208
OPERATORS
For A € a(A) define Zx €
^ = i/r^-^
C-A'
where T is a contour that surrounds a in (cr(A)\cr)c. Clearly, ZAP^ = P<JZ\ which implies that R(Pa) and 01(1 — Pa) are invariant subspaces of Z\. Theo rem 4.2.10 and (4.43) imply (A
P
^~1
\\7 -P - J - [-*--[
ifXGa
If A G a ( A ) \ a and x e R{Pa), then a; = Pax = (Ai - \)Zxx
= Zx{Ai - X)x.
Hence A e p(A\), and by (4.46), cr(A)\a U p(A) C p(A\) and therefore a(Ax) C a.
(4.47)
If A e a and x € E ( l - Pa), then a; = (1 - Pa)x = -(A2 - X)Zxx and Zx(A2-X)x
= (A2-X)Zxx
= -x
for
rr € D(A 2 ).
Hence A G p(A2), and by (4.46), cr U p(A) C p ^ ) which implies a(A2) C a(i4)\a.
(4.48)
If \g(j(Ai)Ucr(A2), then(Ai-X)-1Pa-\-(A2-X)-1{l-P(T) is the resolvent of A; hence A 6 /o(-A). Thus, a (A) C cr(Ax) U c r ^ ) , which together with (4.47) and (4.48) imply that cr(Ai) = a and cr(A2) = cr(A)\a. D Note that in (d) below we have estimates for both negative and positive times. See (4.36) for the definiton of e~Alt when t < 0. Theorem 4.6.7 Suppose that A is a sectorial operator in a Banach space X, that {C e C I R e ( = A} C p{A)
for some
XeR
and that a = {£ (E cr(A) \ Re£ < A} is not empty. Then, a is a spectral set of A. Let Pa be the projection associated with the spectral set a and define A\ = A 1 ^ P I T ) , A2 = Aini_P(T). Then (a) Pae~At
= e-AtPa
fort>0
(b) A1 e »(a(P<,)) and (e-At)WP(r)
= e'A^
fort>0
4.7. THE INHOMOGENEOUS
PROBLEM - PART I
(c) A2 is a sectorial operator in $.(1 — Pa) and (e
209
At
)w(i _ P
A
for t > 0
(d) for some \\ < A < A2 and M < 00 we have that \\e-Alt\\
< Me~Xlt
for
t<0
\\e~A2t\\ < Me~X2t
for
t > 0.
PROOF Since cr{A) is closed and lies in a sector, Lemma 4.5.5 implies that a is a spectral set. Lemma 4.5.5 and Theorem 4.6.6 imply that a(Ax) = {te
a(A) I ReC < Ax},
a(A2) = {( e a (A) \ R e ( > A2}
(4.49)
for some Ai < A < A2. (4.45), Theorem 4.2.8 and (5) of Theorem 4.3.1 imply (a). Theorem 4.6.6 implies that ft(Po is —A\ and thus (b) follows. The same reasoning gives that the generator of {(e~ At ) iK(1 _ P(r) }£>0 is — A2. Hence, Ai is densely defined and since the bound of its resolvent is no larger than the bound of the resolvent of A, we have that A2 is sectorial, which completes the proof of (c). (4.49) and Corollary 4.5.11 imply the bound on e~A2t. Since a(-Ai) = -cr(Ai), (4.49) and Corollary 4.5.11 imply He"^ 1 )*!! < Me for t > 0. Since e - ^ - * ) = e'^^1 by (4.36), we have the rest of (d). Xlt
D
4.7
The Inhomogeneous Problem - Part I
We shall consider the problem of finding a continuous u : [0, r) —► X such that u'(t) + Au(t) = f(t)
for
t e (0, r ) ,
u(0) = u0.
(4.50)
Throughout this Section it is assumed that: (1) X is a Banach space, UQ 6 X (2) —A is the generator of a strongly continuous semigroup {Q(t)}t>o on X (3) 0 < r < 00, / e L((0,t),X) for t e (0,r). The special case when A is a sectorial operator is examined in more detail in Sec tion 6.2.
CHAPTER 4. SEMIGROUPS
210
Theorem 4.7.1 If u G C([0,T),X)
OF LINEAR
OPERATORS
and it satisfies (4.50), then
u(t) = Q(t)u0 + ( Q{tJo
s)f{s)ds
for
t G [0, r ) .
(4.51)
PROOF Choose t G (0,r) and let v(s) = Q(t - s)u(s) for 0 < s < t. If s, s + h G (0, £], h ^ 0, then Theorem 4.3.1 implies
"(«+ *»)-»(«)
=
h
Q('-*-M-Q(*-*)M(S) h
Q(t-s- h) M ^?
i
AQ(* - s)u(s) + Q{t -
=
+ Q (
, _
S
_
W )
+
^ - u'(«)J s)u'(s)
Q(t-s)f(s).
Since lima_>o^(s) = Q{t)uo, Theorem 4.2.11 implies (4.51).
□
So, it would seem that by denning u by (4.51) the problem is solved. However, things are more complicated. First of all, it has been shown that the Bochner integral in (4.51) exists only when (4.50) has a solution. Therefore, we need Lemma 4.7.2 The Bochner integral F(t) = fQ Q(t — s)f(s)ds Moreover, F G C([0,r),X).
exists for all t G [0, r ) .
PROOF Choose t G (0, r) and let / i , /2, • • • be the simple functions approxi mating / G L((0,t),X) as in Corollary 4.2.4. Define 9kt(s) = Q(t - s)fk(s),
gt(s) = Q(t - s)f(s)
for
s G (0,t), k > 1.
It is easy to see that gkt G L((0,£),X) and hence the Dominated Convergence Theorem 4.2.7 implies that gt G L((0,t),X) and J* gkt -> J* gt = F(t). Let gt{s) = 0 for s > t. Hence F(t) = fQT gt(s)ds. If tn -+ t G [0,r), then Ptn(5) ->• 9t(s) for 5 ^ *; hence the DCT implies F(tn) -> F(t). D Define the mild solution of (4.50) to be u given by (4.51). The mild solution may not be differentiable and it may not lie in the domain of A even when / = 0, so it may not be an actual solution. However, if (4.50) has an actual solution, then the actual solution has to be equal to the mild solution by Theorem 4.7.1. The following Theorem 4.7.3 generalizes (3) of Theorem 4.3.1 and shows that the mild solution always satisfies (4.50) on average.
4.7. THE INHOMOGENEOUS
PROBLEM - PART I
211
Theorem 4.7.3 The mild solution u of (4-50) is the unique u G C([0,r),X) which satisfies J0 u(s)ds G ^(A) and u(t) - u0 + A I u(s)ds = / f{s)ds for all t G [0,r). Jo Jo PROOF Let u be given by (4.51). Note that for t G [0, r ) , rt
I u(r)dr Jo
rt
rt
rr
=
/ Q(r)uodr + dr Q(r — s)f(s)ds Jo Jo Jo
=
[ Q{r)u0dr + [ ds [ Jo Jo Js
=
/ Q(r)u0dr+ Jo
rt
rt
Q{r-s)f{s)dr
rt—s
ds Jo
Q{r)f(s)dr. Jo
Hence (3) of Theorem 4.3.1 and Theorem 4.2.10 imply that /0* ueT>(A) A f u(r)dr Jo
= u0 - Q(t)u0 + / (/(s) - Q(t Jo =
(4.52)
and
s)f(s))ds
wo -u(t)
+ / f(s)ds. Jo Therefore the mild solution of (4.50) satisfies (4.52). Let v be a continuous solution of (4.52) and let w(t) = f0(u — v). Then w' + Aw = 0, w(0) = 0 and hence (8) of Theorem 4.3.1 implies that w = 0. □ Corollary 4.7.4 If UQ G D(A) and f(t) = c + f*g for some ce X, g G L((0,t),X) for all t G (0,r) 7 then there exists u in C 1 ([0,r),X) such that (4-50) holds. PROOF
Let v e C([0,r),X) be such that v(t) + AUQ -c + A I v(s)ds = / g(s)ds Jo Jo
and let u(t) = uo + J 0 v.
for all
t G [0, r) □
One cannot weaken the condition on / in the above Corollary to / G C([0, r ) , X), see Exercise 21. Corollary 4.7.5 / / u o £ ^ ( ^ ) a™d / , Af Cl(]fi,T),X) such that (4.50) holds.
G C ( [ 0 , T ) , X ) 7 £/ien there exists u in
PROOF Note that the mild solution u, given by (4.51), lies in T){A) by Theorem 4.2.10. Lemma 4.7.2 implies that Au G C([0,r),X)\ hence A f0 u = Jo Au and we can differentiate (4.52). □
CHAPTER 4. SEMIGROUPS
212
4.8
OF LINEAR
OPERATORS
Exercises
1. Show that Q : [0, oo) —Y Q3(CW(R)), given in Example 4.1.3, is not continuous. 2. Let 7i be a Hilbert space, let {(pi, y?2, • • •} be a complete orthonormal set in % and let Ai, A2,... be complex numbers. Define A by oo
00
jb=i
fc=i
^ / = X>(/,Wb)wb for f eV(A) = {f en \Y^\*k(f,ifk)\2 < 00}. Show that <xp(A) = {Ai, A2,...} and that a (A) = CJP(A). 3. Suppose / G C 2 [0,2TT], /(0) = /(2?r), / x (0) = / X (2TT) and let u satisfy ut = u x x -f (ufx)x
for
0 < re < 27r, £ > 0,
M(0, *) = U(2TT, t), ^ ( 0 , t) = WX(2TT, t)
for
t > 0,
2
tx(-,0)eL (0,27r). Show that there exists c such that lim u(x,t) = c e~ / ( x )
for
0 < x < 2ir.
4. If A is as in Section 4.4, show that there exists 8 G (0,00) such that (Im A)2 < 48(8 + Re A)
for all
A G a (A).
5. Let —A be the generator of a strongly continuous semigroup {Q(t)}t>o on a Banach space X such that the spectrum of A contains a scalar A with Re A < 0. Show that there exists x G T>(A) such that Q(t)x is unbounded for t > 0. 6. Show that the generator of the semigroup in Example 4.1.5 is the operator A defined in Example 4.3.10. (Hint: Compare the resolvents.) 7. Prove the 'Observe ...' statements in Example 4.5.4. 8. Consider n
Au = -Au + ^2 ai®iu + aou
for
u £ ^i-A) = S
1=1
as an operator in WQ,p(Rn), 1 < p < 00. Assume that a* G L°°(E n ) and when p = 00, assume also that a^ are continuous. Show that A is closable and that A is equal to the operator T given in Example 4.5.8.
4.8.
EXERCISES
213
9. Let B be the linear operator in L2(Rn) associated with the sectorial form r
3(u,v) = / where a; G L°°(Rn). when p = 2.
n
J2(DiuDiV + ai(Diu)v) + a0uv
u,v G W1{Rn),
for
Show that P is equal to the operator A given in Exercise 8
10. Suppose that u : (0,T) -» L^ft) has m continuous derivatives where 0 < T < oo, 1 < p < co and Q, is a nonempty open subset of E n . It would be nice to have that if u<p(n)
for
0
then u G W%f{Q x (0,T)). Show that this is not true. 11. Suppose that A e 2t(a, Af,d, A") and C € a{A)\{a}.
Show that Msin<9 > 1.
12. Show that every bounded operator on a complex Banach space is a sectorial operator. 13. Let M, N be closed subspaces of a Banach space X such that X = M + N and Mfl AT = {0}. Show that there exists a projection P on X such that M = ft(P) andiV = # ( l - P ) . 14. Show that a nonempty bounded subset a of a(A) is a spectral set of a linear operator A in a complex Banach space iff there exists S > 0 such that |C - A| > 8 for all
( G a , A G <J(A) \<J.
15. In the proof of Theorem 4.6.6 it is assumed that A is closed. Why is this true? 16. Let v G Z/2(0,7r) be represented as v(x) = Y^=i bnsmnx.
Define
oo
u(x,t) = > j 6 n e _ n ^innrr
for
t > 0, x G (0,7r).
n=l
Show that ||u(-,*)||p < IMIp for p G [l,oo], £ > 0. [Hint: estimate the resolvent in Theorem 4.3.5 by using Exercise 20 in Chapter 1.) 17. Let —A be the generator of a contraction semigroup. Show that Px||2<4||A2x||||a;||
for
x G T>(A2).
Hint: if {Q(t)}t>o is the semigroup and x G D(A 2 ), then tAx = x-
Q(t)x + / (t - s)Q{s)A2x ds Jo
for
t > 0.
214
CHAPTER 4. SEMIGROUPS
OF LINEAR
OPERATORS
18. Use the above problem to improve Lemma 3.4.11 to l l / ' l l p < e | | / % + ll/llp/e
for
/ € C2(R), e e (0,oo), p € [l.oo).
19. Show that if A is a sectorial operator with compact resolvent, then e~At is compact for t > 0. 20. Find a semigroup {Q(t)}t>o such that Q(t) is not compact for any t > 0, yet its generator has a compact resolvent. (Hint: analyze examples given in the Introduction.) 21. Show that the condition on / in Corollary 4.7.4 cannot be weakened to requiring that / e C ( [ 0 , T ) , X ) . (Hint: take f(t) = Q(t)x.)
Chapter 5
Weakly Nonlinear Evolution Equations 5.1
Introduction
The term weakly nonlinear evolution equations will be applied to abstract evo lution equations of the form u'(t) + Au(t) = F(t,u(t))
for
t e [0, r ) ,
(5.1)
where A is a generator of a strongly continuous semigroup in a Banach space X and F is continuous and locally Lipschitz continuous in the last variable. There are no restrictions on A; hence both hyperbolic, including the wave equation, and parabolic problems can be treated. However, the restriction on F can be quite severe. Hence the term weakly nonlinear evolution equations is used. The restriction on F is not so bad when one can choose to work in various spaces of continuous functions. For example, equations like ut = Au + eu,
ut = Au+
1—u are studied in Section 5.3 in a closed subspace of Cs(M n ). Basic existence, uniqueness, continuous dependence and stability results for weakly nonlinear evolution equations are presented in this chapter. The basic ideas used to prove these results are the same as in the ODE theory and are also the same as in the next chapter where semilinear parabolic problems are studied. However, technical preliminaries are much simpler here than in the next chapter. Most of the chapter is devoted to studying approximations of solutions of such equations. By further extending these results, one can actually remove the condition that F is locally Lipschitz continuous in X (Example 5.6.2). The following version of Gronwall's inequality will be sufficient here. 215
216
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
Theorem 5.1.1 (Gronwall) If £ G (0, oo) and f G L 1 (0,^) is real valued such that f(x)
+b
f(s)ds
for
x G (0,^) a.e.,
Jo where a G E and b G [0, oo), then f(x) < aebx for almost all x in (0,^). PROOF If F(x) = be~bx fQx f{s)ds, then F'(x) < abe~bx which implies that bx a + e F{x) < aebx. O The key to all basic results presented in the next section is the following Theorem. Note that for (5.2) to be satisfied we must have that J 0 u G 1){A). T h e o r e m 5.1.2 Suppose —A is the generator of a strongly continuous semigroup {Q(t)}t>o on a Banach space X, a G R, M G R, \\Q(t)\\ < Me~at for t > 0, 0 < r < oo, H : [0, r] x X -» X is continuous and for some L < oo, \\H(t,x)-H{t,y)\\
for
0
Then for each UQ G X there exists a unique u G C([0, r],X) which satisfies u{t)-u0
+A
u(s)ds= / H(s,u{s))ds for all te[0,r]. Jo Jo Moreover, if G : X —> C([0, r],X) zs ^z'ven 6i/ G(UQ) = u, then \\G(x)(t)-G(y){t)\\<M\\x-y\\e(ML-^
for
(5.2)
0
(5.3)
PROOF Abbreviate Y — C([0, T],X) and note that Y is a Banach space with norm || • H^ (Example 1.3.2). Define T : Y -► y by (Tti)(t) = Q(t)w0 + I Q(t~ s)H(s,u{s)) liu,v
ds
for
ueY,0
G y , then ||(Ttx)(t) - (Tv)(t)\\
v^
where c = s u p 0 < t < T ||Q(t)|| and hence ||(T»«)(t) - (T»«)(t)|| < ^ ^ l l « - «lloo
(5.4)
for n > 1, 0 < t < r. Thus, the Contraction Mapping Theorem 1.1.3 implies the existence of a unique u G Y such that Tu = u. Theorem 4.7.3 and closedness of A imply (5.2).
5.1.
INTRODUCTION
217
Choose x, y G X and let u = G(x),v = G(y). For 0 < t < r we have
\\u(t)-v(t)\\
< Me-at\\x-y\\-^ML
e-a^-^\\u{s)-v(s)\\ds
[ Jo
eat\\u{t) - v{t)\\
< M\\x - y\\ +ML
f eas\\u(s) - v{s)\\ds; Jo
hence, the Gronwall Theorem 5.1.1 implies (5.3).
□
The following Corollary says that bounded perturbations of generators of strongly continuous semigroups are still generators of strongly continuous semigroups. This is an important result, but should not be surprising. However, it is interesting that it is a natural consequence of Theorem 5.1.2 which deals with nonlinear problems. Corollary 5.1.3 Suppose that —A is the generator of a strongly continuous semi group on a Banach space X and that B G %$(X). Then —A + B is the generator of a strongly continuous semigroup on X. PROOF Theorem 5.1.2 implies that for each x G X there exists a unique u G C([0,oo),X) such that u(t) = x + {-A + B) [ u(s)ds
for all
Jo
t G [0, oo).
(5.5)
Define R(t)x = u(t) for t > 0. Uniqueness of it and formula (5.5) imply linearity of R{t) for t > 0. (5.3) implies that R(t) G <8(X) for t > 0. If t , r > 0, then obviously t+r
/
u{s)ds]
hence R(t)R{r)x = R(t)u(r) = u(t+r) = R{t-\-r)x. Thus {R{t)}t>o is a strongly continuous semigroup on X. Let —S be its generator. Note that Theorem 4.3.2 implies that (-oo,a) G p(A - B) D p(S) for some a eR. If x G T*(S) and u is as in (5.5), then 1 fh - / u(s)ds ->x h Jo
1 fh 1 and (A - B)~ / u{s)ds = -{x - R(h)x) -* Sx h J0 h
as h -> 0+ and hence the closedness of A - B implies that A - B is an extension of S. Lemma 1.6.14 implies that A - B = S. □ Rr given below is called a retraction map. It will enable us to apply Theo rem 5.1.2 to a locally Lipschitz continuous F.
218
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
Theorem 5.1.4 If X a Banach space, r > 0 and Rr : X —>• X is given by Rr(x) = x when \\x\\ < r then \\Rr(x) - Rr(y)\\ <2\\x-y\\ PROOF
and
for
Rr(x) = j^x
when \\x\\ > r,
allx,yeX.
Let B = {x G X | ||a;|| < r}. If x,y G B, there is nothing to prove.
If x G B and y G X\B,
then T X-y+(l-rr-rr)y
Rx-Ry
=
\\Rx-Ry\\
<
\\x-y\\
<
2||x-y|| + W - r < 2 | | a r - y | | .
If x, y e X\B,
+
\\y\\-r
then Rx-R K
\\Rx-Ry\\
y
~
\\x\\[
V)
llxll"- 11
<
||a:-y|| + | | M | - N | | < 2 | | a ; - y | | .
^
V
a
5.2
Basic Theory
Throughout this section it is assumed that (1) X is a Banach space. (2) —A is the generator of a strongly continuous semigroup {Q(t)}t>o on X. Let M e [1, oo), a E E be such that ||<2(*)|| < Me~at for t > 0. (3) T G (0, oo], U is an open set in X, F : [0,T) x U ->> X is continuous and for each t G (0,T) and each z eU there exist £ > 0 and L < oo such that ||F(S,z)-F(s,y)||
for
z,y e £ ( M ) , a € [0,*].
For r G (0,T], let Sm(r) be the collection of all u G C([0,T),ZY) which satisfy flu{s)ds G D(,4) and u(t) - u(0) +A I u{s)ds = / F(s,u(s))ds for all t G [0,r). Vo Jo Recall that Theorem 4.7.3 implies that u G S m (r) iff u G C([0,r),ZY) and
(5.6)
u(t) = Q(t)u(0) + [ Q(t- s)F(s,u(s))ds ./o So S m (r) is the set of mild solutions of (5.1).
(5.7)
for
t G [0, r ) .
5.2. BASIC
THEORY
219
L e m m a 5.2.1 Suppose r G (0,T) and w G C([0,r],U). L < oo and a continuous H : [0, r] x X —»• JC swc/i £/ia£ (%) i / t e [0,r] and z G B(w{t),8),
then z eU and H(t,z)
(ii) \\H(t,x) - H(t,y)\\ < L\\x -y\\ for t e[0,r], PROOF
Then there exist 5 > 0, =
F(t,z)
x,y e X.
For t G [0,r] choose 8(t) > 0 and L(i) < oo such that
\\F(s,x)-F(s,y)\\
for
x,y e B{w(t),6(t)),
s e[Q,r]
and let fi(t) > 0 be such that \\w{s) - w(t)\\ < S(t)/2 if \s - t\ < /i(t). Com pactness of [0, r] implies that [0,r] C B(ti,fj,(ti)) U • • ■ U B(tn,/j,(tn)) for some U G [0,r]. Let L = 2max;L(£;) and (5 = min*£(^)/4. Note that if t G [0, r] and z,y G B{w(t),2S), then x,|/GZi and ||F(t,a;) - F(t,y)\\ < L\\x - y||/2. Define #(£,x) = F{t,w(t) + i^(rr - w(t))) for t G [0,r] and a; G X, where R§ is the retraction map as given in Theorem 5.1.4. □ Existence for the initial value problem: T h e o r e m 5.2.2 For each uo G U there exist 0 G (0,T) and u G S m (0) such that u(0) = uo. Furthermore, liuin-^oosup0
= Q(t)u0 + [ Q{t-s)F{s,uk(s))ds Jo
for
t e [ O , 0 ] , A; = 1,2,...
PROOF Pick r G (0,T) and let w(t) = u0 for t G [0,r]. Let 8 and H be as given in Lemma 5.2.1 and let u G C([0,T],X) be as given by Theorem 5.1.2. Continuity of u implies that for some 0 G (0, r] we have for t G [0,6] that \\u(t) -w(t)\\ < 6. Hence H(t,u(t)) = F(t,u(t)) and therefore u G S m (0). If 0 is small enough, then (5.4) implies the 'Furthermore' part. □ Uniqueness for the initial value problem: T h e o r e m 5.2.3 IfO
$m(n), u G S m (0) and w(0) = u(0), then
PROOF Choose any r G (0, fi) and let 5 and if be as given in Lemma 5.2.1. For some T' G (0,T] we have \\u(t) - w(t)\\ < 6 for t G [0,r'] and hence H(t,u{t)) = F(t,u(t)). Theorem 5.1.2 implies that w = iu on [0, r'], and hence we can choose T' = T. □
220
CHAPTER
5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
Continuous dependence on the initial value: Theorem 5.2.4 Suppose 0 < r < // < T and w G Sm(fj,). Then there exist e>0 and C < oo such that for every x G B(w(0),e) there exists u G Sm(r) such that u(0) = x and \\u{t) - w(t)\\ < C\\u(0) - w{0)\\ for all t £ [0,r). (5.8) PROOF Let 5, L and if be as in Lemma 5.2.1. Let C = max{Me( M L - a ) T , M } and choose e G (0,5/C). Suppose x G B(w(0),e). Let u = G(x), where G is as in Theorem 5.1.2 and note that (5.8) holds. Since Ce < 5, Lemma 5.2.1 implies that H(t,u(t)) = F(t,u(t)) for t G [0,r] and hence Solutions of real equations, with real initial conditions, are real: Theorem 5.2.5 Suppose that Y is a closed real subspace of X such that (1) (A - X)-XY C Y for some
\
(2) F{t,x) G Y when t G [0,T)
andx€YC\U.
Suppose also that u G Sm(0) for some 6 G (0,T] and that u(0) G Y. u(t)EY
Then
for all *G[O,0).
PROOF Choose any r G [0,6) such that u(t) G Y for t G [0, r). Theorem 5.2.2 implies that there exist 8 G (0,6 — r) and v G C([0, J), U) such that v(t) - U(T) + A / v(s)ds = J0
F(T + s, u(s))ds
for all
t G [0, (5)
Jo
and that t> = limn-^ooVn for some v n G C([Q,5),U) given by ^i(t) = U(T) and Vn+iW = Q(t)n(r) + I Q(tJo
S)F{T + 5, u n ( 5 ))d5
for t G [0, S) and n = 1,2,... Since u{r) G Y, Theorem 4.6.1 and Corollary 4.2.5 imply that the range of every vn is in Y. Hence, the range of v is in Y. l>ztw = u on [0,r] and w(t) = v(t - r) for te[r,r + S). It is clear that w G S m (r + 6). The uniqueness Theorem 5.2.3 implies that w = u on [0,r + (5). Hence, w(t) G Y for £ G [0, r + 5). Therefore, the supremum of such r has to be equal to 6. □ Maximal interval of existence:
5.2. BASIC
THEORY
221
Theorem 5.2.6 Suppose UQ E U. Then there exist r E (0,T], u E S m (r) such that u(0) = uo, and which also have the property that r > \x for all /i E (0,T] for which there exists v E Sm(A0 with v(0) = UQ. Moreover, if r < T, then either JQ ||F(t, ii(t))||dt = oo or there exists x E U\U such that \imt-+T u(t) = x. PROOF Theorems 5.2.2 and 5.2.3 imply the first assertion. Suppose r < T and JQ \\F(t,u(t))\\dt < oo. The Dominated Convergence Theorem 4.2.7 and (5.7) imply that there exists x E X such that \imt-+Tu(t) — x. Clearly x E U. The proof will be complete if we show that the assumption x E U leads to a contradiction. So, assume x E U, choose r' E (T, T) and let u(t) = x for t E [r,r']. Let S and H be as given in Lemma 5.2.1 applied to u E C([0,T'],U). Theorem 5.1.2 implies the existence of v E C ( [ 0 , T ' ] , X ) such that v{t)-u0
+A
Jo
v{s)ds=
/ H(s,v(s))ds
Jo
for
*E[0,r']
and its uniqueness implies that v = uon [0, r ) . For some \i E (r, r r ] we have that v(t) E £(u(t),(J) for t E [0,/x] and hence H{t,v{t)) = F{t,v(t)) for t E [0,/i]. Therefore v E Sm(A0> which is a contradiction to the maximality of r. □ Stability: Theorem 5.2.7 Suppose a > 0, e E [0,a/M], T = oo, 6 > 0, B{0,6) C ^ and \\F(t,x)\\<e\\x\\
for
t>Q,xeB{0,6).
Then for each x 6 5(0, S/M) there exists u E S m (oo) such that u(0) = x and \\u(t)\\ < M\\x\\e^M£-a^
for
t>0.
(5.9)
PROOF Pick the maximal r for which there exists u E Sm(r) satisfying u(0) = x (Theorem 5.2.6). Note that for some fi E (0, r] we have that \\u(t)\\ < 5 for 0 < t < fi. Let r' be the largest of such /i. (5.7) implies \\u(t)\\ eat\\u(t)\\
< Me-at\\x\\+Me
[ Jo
e-a{t-s)\\u(s)\\ds
f eas\\u(s)\\ds for 0 < t < r' Jo and therefore the Gronwall Theorem 5.1.1 implies the bound (5.9) for t E [0, r'). If T' < r, then (5.9) implies that ||U(T')|| < M\\x\\ < 5, which contradicts the maximality of r'. Hence r' = r. If r < oo, then obviously f£ \\F(t,u(t))\\dt < oo. Hence Theorem 5.2.6 implies that \imt^yTu(t) = y E U\U, which is not possible because ||y|| < M\\x\\ < S. Therefore r = oo. □ < M\\x\\+Me
222
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
5.3
Example: Nonlinear Heat Equation
EQUATIONS
We shall discuss the initial value problem for ut (re, t) = Au(x, t) + f{u(x, t)) U(X,0)=UQ(X)
i6En,t>0
for
for
iGf,
l
where / E C ((a, 6),K) for some —oo < a < b < oo. The problem will be set in the real Banach space Ce- Let A^ be the generator of the contraction semigroup on Ce as given in Example 4.5.4 - but restricted to real functions (Example 4.6.3). Observe that A^ is equal to the closure in Ct of the operator A, with the domain consisting of functions in the form c + v where c E l and v is a real valued, rapidly decreasing function. Let U consist of those v E Ct such that V- = inf v(x) > a
and
v+ = supv(rr) < b.
X
X
If v eU, then V- — e > a and v+ + e < b for some e > 0; hence ll/W-/Hlloo<^lk-^||oo
for
z,weB(v,e)
where L = max t; __ c < a ;< t , + + e \f'(x)\. Assume that UQ E U. Let r E (0, oo] and u E Sm(^") be as given by Theorem 5.2.6. Theorem 5.2.3 implies that for every \x E (0,r] this u is the unique element of C([0, fJ>),Ct) which satisfies fQ u(s)ds E X>(A^) and u(t) -u0
= Ae
u{s)ds + / f(u{s))ds for all t E [0,//). (5.10) Jo Jo So ^ is the solution of the original problem on average. By showing its regularity, one can show, see Example 6.4.1 in the next chapter, that it is also the unique actual solution. (5.10) obviously implies that if r E [0, r) and u(t) — u(t + r) for t E [0,r — r), then u E S m (r — r). Theorem 5.2.3 implies that this u is the unique solution starting at u(r). Thus we have the semigroup property. Theorem 5.2.6 also implies that if the solution does not exist for all time, i.e. r < oo, then either JQr sup x \f(u(x,t))\dt = oo or approaches to U\U, the boundary of U. When, for example, f(u) = ± e u , then the solution either exists for all time or else sup 0< £< T ||u(*)||oo = °°- When, for example, / is bounded and (a, 6) = E, then the solution exists for all time. Let h(t) E R be the value of u(t) at infinity, i.e. h(t) = £{u(t)) where I E Q * is as defined in Example 4.5.4. The fact that l{E±iv) = 0 for all v E D(A^), when applied to (5.10), implies that h satisfies an ODE, ti(t) = f(h(t))
for
*E[0,r).
5.3. EXAMPLE: NONLINEAR
HEAT
EQUATION
223
When uo is a constant function, then the uniqueness of u implies that u(x, t) = h(t) for all x G R and t G [0, r ) . When, for example, f(u) = l/(l — u),b = — oo and a = 1, then the solution of the ODE is h(t) = 1 - y/{l - h{0))2 - 2t, which implies that r < (1 — h(0))2/2. T > (1 — supxuo(x))2/2 for this / .
The following Theorem 5.3.1 implies that
Theorem 5.3.1 If v,w G S m (ji), w(0) > v{0) and f 0 < t <
> 0, then w(t) > v(t) for
fi.
PROOF Suppose that r G [0,/i) and that w > v on [0,r]. The proof will be complete if we show that there exists 6 > 0 such that w > v on [0, r + 6]. By Theorem 5.2.2 we can choose v^ G C([0,6V],U) such that vi(t) = v(r), vk+1(t)
= e^vir)
+ [ eA^-s^f(vk(s))ds Jo
for
t G [O,0V], k = 1,2,...
and £fc converge to some v G Sm(^v) with v(0) = v(r). Here 0V G (0, fd — r). The semigroup property implies that v(t) = v(r +1) for t G [0,0V). Choose Wk G C([0, 0iy],W) in the same way and let 0 = mm{0v,Ow}
> 0.
In view of the integral representation of eAtt (Exercise 4), we conclude that eAtt p r e s e r v e s positivity. Hence eAet{w{r) - v{r)) > 0 for
t > 0.
If Wk > Vk on [0,0], then using fDfc+i(t) - vM(t)
= eA<\w(r) - v(r)) + / " c A ^*-)(/(ti) fc ( a )) - /(tfc00))
and the fact that pointwise evaluations are bounded linear functionals on Ci gives that Wk+i > £fc+i ° n [0,0]. Since Wk,Vk converge to w,v in Ci, we have that w >v on [O,r + 0]. □ We have the same stability result as for ODEs: Theorem 5.3.2 If f(c) = 0 and /'(c) < 0 for some c G (a, 6), tfiera /or eac/i /x G (0, —/'(c)) tfiere ezzste £ > 0 swc/i £/ia£ «/ ||^o - c||oo < 8, then u G S m (o°) a™d ||t*(t) - c||oo < ||uo - cllooc-^* for
t>0.
224
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
PROOF Let A = -A£ - /'(c) and F{v) = f{c + v) - f'(c)v. Note that \\e~At\\ < ef'W for t > 0 and that for e = - / ' ( c ) -/x there exists 5 > 0 such that II^MHoo < elMloo if IMIoo < S. Theorem 5.2.7 applies, giving v e S m (oo) A ,F such that v{0) = u0 - c and |M*)lloo < lk(0)||ooe"^. Using (5.6), it is easy to see that v + c G S m (°o) and, by uniqueness, v + c = u. □ As for ODEs, we also have instability when /'(c) > 0. The tools needed to prove this are developed in the next chapter. See Example 6.4.13.
5.4
Approximation for Evolution Equations
We shall approximate the solution u of u\t) + Au{t) = H{t, u(t)),
u(0) = x,
(5.11)
without using any a priori information about u. The original problem is set in a Banach space X. We shall seek approximations in spaces Xn, which are usually finite dimensional spaces in applications and hence this step is sometimes referred to as space discretization. The approximation un will satisfy < ( * ) + Anun(t)
= PnH{t,Enun(t)),
un{0) = Pnx,
(5.12)
where An is a bounded operator on Xn approximating A, Pn denotes a 'projection' from X to Xn and En is an 'interpolation' that associates with each element of Xn an element of X. Using some minimal assumptions, it will be shown that Enun —> u. (5.12) is typically an ODE which has to be integrated numerically. In actual numerical applications it is usually preferred to discretize time t at the same time as space X. One popular way to do this is to use the explicit Euler method, un(t + rn)-un(t)
+
w
PnH(tEn-n(t))
So, if v* denotes the approximation of n n (/cr n ), then vkn+1 = (1 - rnAn)vkn + TnPnH(krn, vkn),
v°n = Pnx.
(5.13)
A variant of the implicit Euler method is
w:
= (1 + enAn)-lw^
+ enPnH (ken, u;*),
w°n = Pnx.
(5.14)
We shall also prove convergence of both Env\i'Tn* and EnWn to u(t). The following assumptions about space discretization will be used: (A)
(a) X, X\, X 2 , . . . are all real or all complex Banach spaces. All norms will be denoted by || • ||.
5.4. APPROXIMATION
FOR EVOLUTION
EQUATIONS
225
(b) Pn G 1, x G X. (c) En G *B(X n ,X), q G [0,oo) are such that \\Enx\\ < q\\x\\ for n > 1, x G X n . (d) PnEnx
= x for n > 1,
About A n it will be assumed that (B) An G 2$(-X"n) are such that for some M < oo, a G 1R we have ||e-^n*|| <
Me-at
f()r
t
> 0 , n >
L
This is called the stability condition of approximations An. While all norms are equivalent in finite dimensional spaces, the equivalence is in general not uniform in n. So the norms on spaces Xn have to be chosen carefully in order to insure that the bound in B does not depend on n. Observe that B and Theorem 4.3.2 imply that (—oo,a) C p{An) for n > 1. A bit more restrictive condition will be needed to prove convergence when the explicit Euler method is used. The only connection between A and An is represented by (C) A is a densely defined linear operator in X and there exists Ao G (—oo, a)f)p(A) such that for all x in a dense subset of X, lim \\En(An - Xor'PnX
X0)-lx\\
-(A-
= 0.
n—^oo
Note that C requires that the approximations An can be used to solve the time independent problem, Au — \QU = x, for all £ in a dense subset of X. It turns out that this formulation of assumptions is especially convenient to apply to the Galerkin method of approximating solutions of parabolic and wave type equations (Sections 5.6, 5.7, 5.8). For finite difference ap proximations (Section 5.5), it can be more convenient to verify the following condition instead. ( C ) A is a densely defined linear operator in X, Ao G (—oo,a) D p{A) and lim \\AnPnu - PnAu\\ = lim \\EnPnu - u\\ = 0 for all
n-»oo
n-^oo
u G D(A).
Lemma 5.4.1 Assume A and B . Then, C implies C. Choose x G X and let u = (A - XQ)~1X.
PROOF
- \ )-lP x
E {A n
n
0
n
-u = E P u -u + E (A n n
n
n
Note that
- A ) _ 1 (Pn^^ 0
hence B, the bound in Theorem 4.3.2 and C imply C.
A P u), n n
□
226
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
The following is a version of the Trotter Theorem. Theorem 5.4.2 Assume A, B and C. Then —A is the generator of a strongly con tinuous semigroup {Q(t)}t>o on X. Moreover, ||Q(£)|| < pqMe~at for t > 0 and lim snpebt\\Ene~AntPnx PROOF implies
- Q(t)x\\ = 0 for all xeX.be
(-oo,a).
Abbreviate Rn(X) = ( A n - A ) _ 1 andR(X) = {A-X)'1. \\Rn{X)m\\ < M{a - X)-m
This implies that lin^^oo EnRn(Xo)Pnx
for
Theorem 4.3.2
A < a, m > 1, n > 1.
(5.15)
= R(Xo)x for every x e X.
If A e (-oo, a) fl p(A), then EnRn(X)Pn-R(X)
=
(1 + (A - X0)EnRn(X)Pn)(EnRn(Xo)Pn
- R(XQ))(A
-
X0)R(X).
Therefore, if A e (-oo, a) D p(A), then lim EnRn(X)Pnx
= R(X)x for a; € X.
(5.16)
n—>-oo
(5.16) and (5.15) imply if A 6 (-00,0) np(A), then ||JR(A)||
(5.17)
If fi e (—00, a) and /i 0 p(A), then ||.R(A)|| should approach +00 as A goes from Ao to /i. However, by (5.17) this is not possible. Therefore, (—00,a) C p{A). Induction on m gives lim \\EnRn(X)mPnx
- R(X)mx\\ =0
for
x e X, m > 1, A < a.
(5.18)
n—¥GG
This and (5.15) imply that P ( A ) m | | < pqM(a - X)-m
for
m > 1, A < a.
Therefore, Theorem 4.3.5 implies that —A is the generator of a strongly contin uous semigroup {Q{t)}t>o with the stated bound. Choose now A < a , 6 < a, x e X. Since ^-e-^^ 5 ) J R n (A)P n Q( 5 )i?(A)a; = e-A^-^Pn{R(X)
-
EnRn(X)Pn)Q(s)x1
5.4. APPROXIMATION
FOR EVOLUTION
EQUATIONS
227
we have that e 6 t ||E n i^(A)(P n Q(t) -
e-A-tPn)R(X)x\\
= II f eW-^Ene-^-'lPniRiX) \\Jo roo
< pqM / Jo
EnRn(X)Pn)ebsQ(s)xds
-
EnRn(X)Pn)ebsQ(s)x\\ds
\\(R(X) -
and the DCT implies lim supe 6 < ||E n ^ n (A)(P n QW - e-AntPn)R{X)x\\
= 0.
(5.19)
n-»oo t > 0
Note that ebt(EnRn(X)PnQ(t) - Q(t)R(X))R(X)x = (EnRn(X)Pn R(X))ebtQ(t)R(X)x = (EnRn(X)Pn-R(X))R(X)x +■/ [\Enl Jo Jo
{EnRn{X)Pn-R{X))e0sQ{s){b-A)R{X)xds.
Hence the DCT and (5.19) imply that lim snpebt\\{Q(t)R(X)
- EnRn(X)e~AntPn)R(X)x\\
= 0.
(5.20)
n->oo t > 0
Since EnRn(X)e~AntPnR(X)x Ant = Ene- Pn(EnRn(X)Pn
Ene-AntPnR(X)2x - R(X))R(X)x,
we see that (5.20) implies lim supebt\\(Q(t)
- Ene-AntPn)R(X)2x\\
= 0.
n-*x> t > 0
We are done because the range of R{X)2 is equal to T)(A2), which is dense in X by Theorem 4.3.1. □ For the explicit Euler method we will need a bit stronger assumption: (B') rn G (0, oo), An G ?B(Xn) are such that for some M < oo, a G E we have ||(1 - rnAn)k\\ moreover, lim^oo ^% — 0.
< Me~~akTn
for
n > 1, k > 0;
228
CHAPTER
5. WEAKLY NONLINEAR
EVOLUTION
Corollary 5.4.3 B' implies B if a < (1 - e~Tnh)/rn
EQUATIONS
forn>l.
This Corollary follows from the following Lemma Lemma 5.4.4 Suppose that X is a Banach space and that A G 55(X) is such that 11(1 - rA)k\\ < Me~~akr
for
k>0
and M G [l,oo). Let a < (1 - e _ d r ) / r . Then
for some r G (0,oo), a£R
\\e~At\\ < Me~at
for
t>0
and ||(1 - rAf'^x PROOF
- e~Mx\\ < rM3e-at+2^T(t\\A2x\\
+ \\Ax\\)
for
xeX,t>
0.
Let B = 1 - TA. In view of (4.36) we have
e-At
e(B-l)t/r
=
e-t/reBt/r
=
k=0
<Me~at.
< Me-^Y^y^ k=Q
Choose x G X and let e^ = Bkx — e~Akrx for k > 0. Induction implies A;-l
ek = YJBje-A^k-l-^Tel
for
k > 1
j=o
and, since a < a, we have fc-i
llcfcll < M ^ e - ^ e - ^ - ^ ^ H e x H < ArM 2 c- a <*- 1 ) T ||e 1 ||. Note ex=x-
TAX - e~rAx =
Jo
T)e~AtA2xdt,
(t -
||ei||
for
k > 0.
5.4. APPROXIMATION
FOR EVOLUTION
EQUATIONS
229
Thus, if t > 0 and k = [t/r], hence, 0 < t — kr < r, then WB^x
- e-Atx\\ < r 2 /cM 3 e- a ( fc - 1+ ^) T ||A 2 a;|| + R
(5.21)
where R = \\e-AkTx
- e-Atx\\ < M [ e-as\\Ax\\ds Jkr
rMe-at^a\T\\Ax\\.
<
Using this in (5.21) implies the conclusion of the Lemma.
□
Theorem 5.4.5 Assume A, B ' and C. Then, —A is the generator of a strongly continuous semigroup {Q(t)}t>o on X. Moreover, \\Q(t)\\ < pqMe~at for t > 0 and lim supebt\\En(l
- rnAn)^T^Pnx
- Q{t)x\\ = 0 for all xeX.be
(-oo,a).
n->oo t > 0
PROOF We shall prove the statements of the Theorem first for the case when a is replaced with a as given in Corollary 5.4.3. Taking the limit n —)> oo then implies the original conclusion. Abbreviate Rn = (An - A 0 ) _ 1 and R = (A - A 0 ) _ 1 . Take t > 0, n > 1, x e X and note (En(l - TnAn)WT^Pn
- Q(t))R2x
= Iln(t) - I2n(t) + /3n(t),
where hn(t)
=
hn(t)
=
hn(t)
En^l-TnAnfM-e-^RlPnX Eniil-TnAn^-e-^PniEnRlPn-R^X Ant
(Ene- Pn-Q(t))R2x.
=
Theorem 5.4.2 implies that lim supe 6t ||J 3 n(*)|| = 0
for
b < a.
n-Kx> t > 0
Since ||/ 2 n (t)|| < pqMie-^M
+ e-at)\\EnR2nPnx
-
R2x\\,
(5.18) implies lim supe6£||72nWII = 0
for
b < a.
n-)-oo t > 0
Using Lemma 5.4.4 we obtain \\hn(t)\\ < TnqM3e-at+2^T"(t\\A2nRlPnx\\
+
\\AnRlPnx\\)
230
CHAPTER
5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
and, since \\Rn\\ < M/(a - A0), \\AnRn\\ < 1 + |A 0 |M/(a - A0), we have lim supe 6 t ||/i n W|| = 0
for
b < a.
This proves the limit in the Theorem for x e T)(A2), which is dense in X by Theorem 4.3.1, and hence for all x e X. Q Convergence for the implicit Euler method follows directly from convergence for the explicit Euler method: Theorem 5.4.6 Assume A, B, C and that {en}^! lim supe 6 *||£ n (l + enAn)-[t/£n]Pnx
C (0,oo) ; limn_>oo£n = 0. Then
- Q(t)x\\ = 0 for all xeX,b£
(-co,a).
71-+00 t > Q
PROOF
Choose N so that 1 + aen > 0 for n > TV. Define A!n = An(l + enAn)-1
Note 1 — enAn = (1 + enAn)~l; \\{l-enA!n)k\\<M{l
for
n>N.
hence (5.15) implies
+ ena)-k
<Me~~aenk
for
k > 0, n > AT,
where aen < ln(l + aen). Hence B ' is satisfied with A'n, en in place of An, rn for n>N. Since \\An(An - A 0 ) _ 1 || < 1 + |A 0 |M/(a - A0) and (A'n - Ac)" 1 = (An - Ao)"1 + enA2n(An - A 0 )~ 2 (l - \QenAn{An
- AQ)"1)"1
when n is large enough, we see that C is also satisfied with An in place of An. Thus, Theorem 5.4.5 implies the conclusion. □ For simplicity of presentation we shall assume that the nonlinearity satisfies (D) r E (0, oo), H : [0, r] X X —> X is continuous and such that for some L < oo, \\H(t,x)-H{t,y)\\
for
0 < t < r, x,y G X.
Using Lemma 5.2.1 one can handle more general nonlinearity, see the example in the next Section. Note, when —A is the generator of a strongly continuous semigroup on X, then D implies the existence of a unique mild solution of (5.11) on [0, r] for every x £ X, see Theorem 5.1.2. Likewise, (5.12) has a unique mild solution on [0, r]. However, since An are bounded operators, Corollary 4.7.5 implies that the mild solution is differentiable and satisfies (5.12).
5.4. APPROXIMATION
FOR EVOLUTION
EQUATIONS
231
Theorem 5.4.7 Assume A, B , C, D and pick x G X. Let u he the mild solution of (5.11) and let un he the solutions of (5.12). Then lim sup \\Enun(t) - u(t)\\ = 0.
Tl OG
^ 0
PROOF
Note that for t G [0, r], n > 1, we have un{t) = e-A»tPnx+
e-A^-s>>PnH(s,Enu(s))ds
[ Jo
u(t) = Q{t)x + [ Q(tJo
s)H(s,u{s))ds.
(5.22)
Hence u{t) - Enun(t)
Q{t)x-Ene-AntPnx
=
{Q{t-s)-Ene-A^t-s^Pn)H(s,u(s))ds
+ f Jo
+ / Ene-^-^PniHis^is)) Jo
-
H(s,Enun{s)))ds.
Define rn(t) = \\u(t) en=
Enun(t)l
\\Q(t)x-Ene-AntPnx\\,
sup 0
dn(s)=
sup \\(Q(t) - Ene-^P^H^uism
<2C\\H(s,u(s))l
0
where C = pqM(l + e~aT). Note that linin^oo^ = 0 by Theorem 5.4.2, which together with the DCT also implies that fQT dn(s)ds = 0. Since rn(t) < £ n + / dn(s)ds + LC I rn(s)ds Jo Jo
for
t G [0,r], n > 1,
the Gronwall Theorem 5.1.1 implies rn(t) < f en+
/
d n (s)dsj
eLC\
which completes the proof.
□
Theorem 5.4.8 Assume A, B ' ? C, D and pick x G X. Let u he the mild solution of (5.11) and let v^ he given by (5.13). Then lim sup n->-oo
0 < t < r
\\Env^-u(t)\\=0.
232
CHAPTER 5. WEAKLY NONLINEAR
PROOF
Let Bn = 1 - rnAn.
EVOLUTION
EQUATIONS
Observe that for n > 1, k > 1, k-l
vkn = BknPnx + r n £
Bkn-l-ipnH{JTn,
Envn).
j=0
Hence, for t > 0, r[t/Tn}r f[t/Tn]Tn n
Env^
= EnB^PnX
+ / /o Jo
Dn(t, s)H([s/rn]rni
= EnB[n,Tn]'l~[s,Tn]Pn-
where Dn(t,s)
Env^)ds,
Using (5.22) we obtain
u(t)-EnvW^
= Y,Iin(t), 2=1
where hn(t)
=
hn(t)
=
Q(t)x-EnB^Pnx [
Q(t-s)H(s,u(s))ds
J[t/rn}rn r[t/Tn]Tn
hn(t)
=
/ Jo
hn(t)
=
/ Jo
hn(t)
=
hn(t)
=
(Q{t -s)-
Q(t - [s/rn]rn - rn))H(s,
u(s))ds
r[t/Tn]rn
(Q(t - [s/rn]rn - rn) -
f[t/Tn]Tn / Jo f[t/rn]Tn
/ Jo
Define rn(t) = \\u(t) -Envh'Tn% bound of ||Q(*)||, \\EnB^Pn\\,
Dn(t,s))H(s,u(s))ds
Dn(t,s)(H(s,U(s))-H([s/Tn]Tn,u(s)))ds
Dn{t,s){H{[s/Tn)TnMs))-H{[slTn}rn,Env^ Sin = sup 0 < t < r ||/i n (*)|| and let C be the upper ||£T(t,ti(t))||
for
t 6 [0,r], n > 1.
Theorem 5.4.5 implies limn^.oo 8\n = 0. Note that <52n < C2rn -» 0 as n -> oo. Note that Ssn
l)H(s,u(s))\\dS]
hence the strong continuity of Q and the DCT imply lim^oo J 3n = 0.
5.5. EXAMPLE: FINITE DIFFERENCE
METHOD
233
Note that <W< / T sup \\(Q(t) JO
EnB^Pn)H(sMs))\\ds;
0
hence Theorem 5.4.5 and the DCT imply limn_>oo J 4n = 0. Since 85n < C fQT \\H(s,u(s))-H([s/Tn]rn, the DCT imply lim^oo 85n = 0.
u(s))\\ds, the continuity of if and
Since rn(t) < J26in
+ LC /
rn(s)ds,
the Gronwall Theorem 5.1.1 implies 5
r„(t) < $ 3
feeLCr
for
t G [0, r], n > 1,
i=l
which completes the proof.
□
T h e o r e m 5.4.9 Assume A, B , C, D, pzcA; i E l and { e n } ^ C (0, oo) such that limn_>.00 en = 0. Lei w be the mild solution of (5.11) and let w^ be given by (5.1J[). Then lim sup \\Enwl/T^ - u(t)\\ = 0. n->oo0
PROOF This proof is identical to that for Theorem 5.4.8, with the following obvious modifications: replace r n by en, v by wy set new Bn = (1 + enAn)~l and use Theorem 5.4.6 in place of Theorem 5.4.5. □
5.5
Example: Finite Difference Method
We shall investigate the following parabolic equation: ut(x, t) = a(x)uxx(x, t) + F(u{x, t)) for 0 < x < 1, t > 0 tx(0, t) = tx(l, *) = 0 for t > 0 u(x,0) = uo(x) for 0 < x < 1, where a G C([0,1], (0,oo)), F G C 1 ^ , ^ ) , ^(0) = 0 and u0 G X = {/ G C([0,1], E) | /(0) = / ( l ) = 0}.
"I \ J
(5.23)
234
CHAPTER 5. WEAKLY NONLINEAR
If u^i is to approximate u(i8n,kTn), discretization is 7 / f c + 1 — 71*
—-
llk
A- 1lk
= a(2()n)— !
— 9<J/ fc
+ F\un i)
^
n,i
= w
o(^n)
EQUATIONS
6n = ^ - , then an obvious finite difference
Un,0 = Un,n+1 = 0 w
EVOLUTION
for
for
* or
1 < « < n, n > 1, A: > 0
fl > 1, fc > 0
1 < 2 < n, ra > 1.
Assuming only the stability condition, 2rna{i8n) < 52n for
1 < i < n, n > 1,
(5.24)
we shall prove that linear interpolations of i4;- converge to the mild solution u of (5.23). Moreover, it will be shown that the convergence is uniform on any finite time interval that is shorter than the time interval of existence of u. Let || • || be the max norm on X and on Xn = R n , n > 1. Define a 'projection' PnE
for
1 < i < n, n > 1.
For n > 1, define an 'interpolation' £Jn 6 *B(X n ,X) by (Enc)(x) = Ci + ( ^ - t)(ci+i - <*) for * < ^ ; < * + l, 0 < i < n , where c = (ci,...,Cn) T E X n and Co = Cn+i = 0. Note that assumption A of Section 5.4 is satisfied, with p = q = 1. For n > 1 define A n E *B{Xn) by (A n c)i = a(i(J„)—
^
—
for
1 < i < n,
where c = ( c i , . . . , Cn)T E X n and Co = c n +i = 0. Note that ((1 - rnAn)c)i = (1 - 2 T B 2 ^ 1 ) C J + rn^c,+1 u
n
+ u
"n
rna-^^V,
n
hence ||1 - T n A„|| < ma*{|l - 2 r „ ^ | + 2 r t
u
n
n
u
^}. n
Therefore choosing rn > 0 so that they satisfy the stability condition (5.24) implies ||l-rnAn|| < 1
for
n>l.
Thus, assumption B ' of Section 5.4 is satisfied, with M = 1, a = 0. Corollary 5.4.3 implies that B is also satisfied.
5.5. EXAMPLE: FINITE DIFFERENCE
METHOD
235
Define A by Af = -af"
for
/€D(A)E{/€C
2
[0,1]|/,/"EI}.
It is easy to show that 0 G p(A) by showing that (A-lf)(x) = £
(l^kfMdy
+£
for
/
6
X, x € [0,1].
Since the resolvent set is open, we have that A G p(A) when |A| is small. If / G D(A) and n > 1, then for 1 < i < n we have (AnPnf
- PnAf)i
=
^l(Slf"(iSn)
=
^
+ 2f(iSn) - f(iSn + Sn) - f(i5n
/ " ( * » - \t\)(f"(i5n) " J-sn and hence the uniform continuity of / " implies that lim \\AnPnf
- PnAf\\ = 0
- f"(i5n
- 6n))
+ t))dt
for / € V(A).
n—>oo
If / G T>(A) and n > 1, then for i < -jL < i + 1, 0 < i < n we have
(EnPnf-f)(x)
= f(iSn) +
(f--i)(f(i5n+Sn)-f(iSn))-f(x)
= h r'n / n ^'( i|5 "+*) - •'"(i<5"+s))d*ds JO
JO
\\EnPnf-f\\ < siwnv Therefore assumption C of Section 5.4 is satisfied and so is C by Lemma 5.4.1. Theorem 5.4.5 implies that —A is the generator of a contraction semigroup. This was shown independently in Example 4.3.9. Thus, Theorem 5.2.6 applies, giving the mild solution u of (5.23) on the maximal interval of existence [0, rmax). Note, if T~max < °o> then u and F(u) have to blow-up as t —> rmax. Lemma 5.2.1 implies that for any r G (0,r m a x ) we can find a Lipschitz continuous nonlinearity H which agrees with F in a neighborhood of u on [0, r]. Theorem 5.4.8 implies that lim sup \\EnvWT^-u{t)\\=0 n
^°°0
where v% is given by (5.13). Thus, for large enough n, EnVn'Tn* is in the neighborhood of u where F and H agree and therefore vk = i4;- for t G [0, r]. This proves that linear interpolations of vrn\T converge uniformly to u on [0, r\. It is natural to expect that in order to approximate u to a given tolerance it would be optimal to choose about the same size space (Sn) and time (r n ) steps. However, the stability condition (5.24) requires that the time steps be much smaller than the space steps. This is due to the use of the explicit Euler method for space discretization (5.13). If one would choose the implicit version (5.14) instead, then there would be no such restriction (any en G (0, oo), such that limn_^oo £n = 0> would do).
236
5.6
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
Example: Galerkin Method for Parabolic Equations
Here the results of Section 5.4 will be applied to u'{t) + Au(t) = H{t, u{t)),
u(0) = x,
(5.25)
where A is the operator associated with a sectorial form #. This covers most parabolic equations set in a Hilbert space. See Section 3.7. Assume that H is a Hilbert space with an inner product (■, •) and norm || • ||, [•, •] is an inner product o n V C ^ with the corresponding norm denoted by | • | and # is a sectorial sesquilinear form on V such that hypotheses H I , H2, H3 of the section on Sectorial Forms are satisfied. Here, assume in addition that (H4) Vi, V2, • ■ • are finite dimensional subspaces of V such that lim inf \y — z\ = 0 n->ooz€V„
for all y in a dense (in | • | norm) subset of V. Observe that hypothesis H4 is exactly the same one as the one in Theorem 2.8.11, where convergence of approximations of solutions of elliptic problems is proved. H I H4 are assumed to be in effect throughout this Section. We shall first demonstrate how assumptions A, B and C of Section 5.4 can be satisfied without imposing any other conditions. Then a version of Theorem 5.4.7 that is suitable for obtaining Galerkin approximations will be presented. Finally, in Example 5.6.2, it will be shown that the Galerkin approximations can be used to deduce existence of solutions even when the nonlinearity is not locally Lipschitz continuous. Let Xn = Vn be equipped with norm || • ||. Let Pn be the orthogonal projection of % onto Xn. Let En be the identity map from Xn to X = %. Thus, assumption A of Section 5.4 is clearly satisfied, with p = q = 1. Observe that assumptions H I , H2 and H3 remain valid, with the same constants, if we replace both % and V with Xn. Let the operator An 6 *B(Xn) be the operator associated with ^-restricted to Xn. Hence, An is defined by (Any, z) = 5(y, z)
for all
y, z e Xn, n > 1.
(1) of Theorem 2.8.2 and Figure 2.1 imply that IK^n-A)-1!!^
_ *_2
.
for
A < a + M 3 Mf 2 .
Hence, the Hille-Yosida Theorem 4.3.5 implies that \\e~Ant\\ < e -( a + M 3M r 2 )t
for
t > 0, n > 1.
(5.26)
5.6.
EXAMPLE:
GALERKIN
METHOD
FOR PARABOLIC
EQUATIONS
237
T h i s shows t h a t assumption B of Section 5.4 is satisfied. T h e o r e m 2.8.11 implies t h a t lim \{An - \)~lPnx
-(A-
A)_1d = 0
for all
x G X, A < a +
M3Mr2;
n-»oo
hence C of Section 5.4 is also satisfied. Therefore, all results of Section 5.4 are applicable. For example, by using (5.26) t o characterize t h e solution of (5.12), we can restate T h e o r e m 5.4.7 as follows: T h e o r e m 5 . 6 . 1 Assume D of Section 5.4, choose any x G H. and let u denote the mild solution of (5.25). Then for every n > 1 there exists a unique un G C 1 ([0, r ] , V n ) such that for all z G V n we have (un(0),z) = (x,z) and (u'n(t),z)+$(un(t),z)
= (H(t,un(t)),z)
for
*G[0,T].
Moreover, lim sup \\un(t) — u(t)\\ = 0. ->°°o
n
EXAMPLE 5.6.2 We shall first prove convergence of finite element approximations to the following parabolic equation: ut(x,t)
= uxx{x,t)
+ G(u(x,t))
for
0 < x < 1, t > 0
u(0,t)=u(l,t) = 0
for
u(x,Q) = u0(x)
0 < x < 1,
for
t>0
where u0 G L 2 (0,1) is real valued, G G C 1 ^ , ^ ) and G' is bounded. Then it will be shown how the condition that G' is bounded can be removed without losing the existence of the solution and the convergence of approximations. Let U be the real L 2 (0,1) and let V, S, [•, •] be as in Example 2.8.1; hence, H I , H2 and H 3 are satisfied. For n > 1, let Sn = l / ( n + 1) and define Vn C V as follows: v G Vn if and only if v G C[0,1], v(0) — v(l) = 0 , v is linear on [iSn, (i + l)<5n] for each i = 0 , 1 , . . . , n. Lemma 2.11.1 implies that H 4 is satisfied for all y G 1>(A), which is dense in V by Theorem 2.8.2, and therefore we can apply Theorem 5.6.1. Let V> be the hat function as in Section 2.11 and let tpn,i(x) that un can be represented as
=
^ ( ^ / ^ n — *)• Note
n Un(x,
t)=Y,
Cn,i{t)^n,i{x).
(5.27)
t=l
Taking z in Theorem 5.6.1 to be ipnj, 1 < J < n, n, > 1 gives the ODEs, yX,»M / i=l
J
°
^nA^nj^dx
+ y2cn,i{t) i=l
/ J
o
iP'n^x)iP'nj(x)dx
=
Sngnj(cn,.(t)),
238
CHAPTER
5.
WEAKLY
NONLINEAR
EVOLUTION
EQUATIONS
where G I Y^cn,i^n,i(x)
tn9n,j(cn,.) = /
I
^n,j(X)dx
with the initial conditions n
-1
rl
V c n , i ( 0 ) / i/jn,i(x)ipnj(x)dx which completely determine c j(t)
= 6nbnJ=
u0(x)ipnj(x)dx
for t > 0. Evaluation of those integrals gives
n
K j
+ C'nj+l + C'n,j-l
+
2cn,j - C^+i - Cn>J-i
S2
6
=
^ ^
}
^
^
4Cn,j(0) + Cn>j+1(0) + Cn,j-i(0) _ ^
6 for 1 < j < n, n > 1, where cn>o = cn)Tl-f i = 0 and 0n,j(Crv) =
/
G(cnJ-illj(x
"
nJ
+ 1) + Cn,j1p(x) + C n , j + i^(x - l))^(x)dx
^o(j^n + ^ ^ / - i
x6n)ip(x)dx.
After solving (5.28) we obtain it n via (5.27), and Theorem 5.6.1 implies convergence of un to the mild solution u on [0,co). Note that it is numerically more difficult to solve (5.28) than the corresponding finite difference scheme in Section 5.5. However, the approach used here is directly applicable to a much larger class of parabolic problems. Note also that, here, the initial value UQ can be any L2 function. Let us remove the condition that G' be bounded from now on. Note that G may not be even locally Lipschitz continuous in L2 and hence nothing in this chapter may seem applicable. However, observe that one can still solve the ODEs (5.28) and hence obtain Galerkin approximations un on at least some finite time interval. Now suppose that one can show that 7=
sup
\un(x,t)\
< oo
(5.29)
0<xl
for some r > 0. Choose <j> G C 1 (E) such that (j){x) = x for \x\ < 7 and (j){x) = 0 for |x| > 7 + 1. Define G(x) = G{(j){x)) for x e E and note that G € C1 (E, E) and G' is bounded. So we can apply the above theory with G in place of G and obtain un converging to the mild solution u. However, (5.29) implies that c n) . also satisfy the modified (5.28) and hence, by the uniqueness for ODEs, cn,- = c n ,. on [0,T] and therefore un = un on [0,r]. Thus, un converge to u on [0,r] and u can be interpreted as the mild solution on [0, r] of the original problem. Thus, when G' is not bounded, everything depends on being able to establish the bound (5.29). In some cases numerical evidence may be acceptable. It will be shown
5.7. EXAMPLE: GALERKIN METHOD FOR THE WAVE EQUATION
239
now that bound (5.29) can always be satisfied for some small r > 0 when u0 e L°°(0,1). To do this we will be using the max norm || • ||oo on Kn. Define an n x n matrix Bn by Bn(i,i) = 2, B„(»,t-1) = - 1 , 5 n (i,t + l) = - l . Note that (5.28) can be written as (1 - \Bn)c'n^ + 8-2Bncn, = gn,{cn,). A long but elementary calculation (Exercise 5) gives that ||(1 -\Bnyl\\<
3 for n > l .
(5.30)
Hence < t . + Dncn, = (1 - \Bn)-lgn,.{cn,), where Dn — S~2(l - ^Bn)~lBn. It can be shown (Exercise 6) that
(5.31)
He-^H < 2 for t > 0, n > 1. This, (5.30) and (5.31) imply that cn,(t) = e-^ £ c n ,.(0) + f e~D^-s\\ Jo
-
\Bn)-lgn,{cn,(s))ds
l|cn,Wlloc < 211^.(0)1100 + 6 / \\gn,(cn,(s))\\oods. Jo Since cn,.(0) = (1 - \Bn)-lbny. and ||6n).||oo < ||u0||co we have l|Cn,.Wlloo < 6||u 0 ||oo + 6 /
||^n,.(c„,.(*))||ood«.
(5-32)
./o Pick any 7 > 6||uo||oo and let G1 = max|x|<7 |G(rc)|. (5.32) implies that l|Cn,-Wlloo<6||u0||oo+6G7*
as long as ||cn).(s)||00 < 7 for 0 < s < t. Hence, if r is such that 6IKII00 + 6G 7 r = 7, then bound (5.29) holds and the Galerkin approximations un converge on [0,r] to the mild solution of the original problem.
5.7
Example: Galerkin Method for t h e Wave Equation
Here the results of Section 5.4 will be applied to u"(t) + Au{t) = /(t,ti(t),u'(t)),
u(0) = 1x0,1/(0) = v 0 ,
(5-33)
where A is the operator associated with a sectorial form J. Assume that Ti is a Hilbert space, with an inner product (•, •) and norm || • ||, [•, •] is an inner product o n V c H with its corresponding norm denoted by | • | and $ is a sectorial sesquilinear form on V such that hypotheses H I , H2 on H3 of the section on Sectorial Forms are satisfied. Here it is also assumed that
240
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
(H4) Vi, V2, • ■ • are finite dimensional subspaces of V such that lim inf \y — z\ = 0 for all y in a dense (in | - | norm) subset of V. These assumptions about %, V, Vn, 5 are the same as in Section 5.6, where the Galerkin method for parabolic equations was studied. In order for the wave equation to make sense we also assume, as in Section 4.4, that there exists b < 00 such that \3(x,y)-3{v,x)\<2b\x\\\y\\
for all
x,yGV.
Assume, in addition, that r G (0,00), / : [0, r] x V x % —> % is continuous and such that for some L < 00, ||/(t,xi,2/i) - / ( t , X 2 , y 2 ) | | < L(\Xl -x2\
+ ||yi - y2\\)
for all 0 < t < r, xi G V, yi G %. In applications one can, as in Example 5.6.2, considerably weaken this condition on / . Theorem 5.7.1 Choose any UQ G V, vo G Ti. Then for each n > 1 there exists a unique un G C 2 ([0,r], Vn) such that for all z G Vn we have K(t),z)+$(un(t),z) [un(0),z]
= (f(t,un(t),u'n(t)),z) = [U0,Z],
(u'n{0),z)
for =
*G[0,r]
(VQ,Z).
Moreover, there exists u G C ( [ 0 , T ] , V) fl C^QO, r],H) such that lim sup (\un(t) - u(t)\ + \\u'n(t) - u'(t)\\) = 0, n
^°°0
u(0) = u0,
it'(O) = v 0 ,
(U(.),*)GC2([0,T],C)
for
z G V,
2
d ^ 2 -(u(t),^)+5(^(t),2;) = (/(t, W (t), W , (t)),2:) PROOF
/or
t € [ 0 , r ] , z G V.
Let Xn = Vn x Vn. Xn is a Hilbert space with an inner product ({x, y}, {z, w}) = [x, z] + (y, w).
Let this also be the inner product on X = V x %. Let P„ be the orthogonal (in V) projection of V onto Vn. Let Pfl be the orthogonal (in H) projection of H onto Vn.
5.7. EXAMPLE: GALERKIN METHOD FOR THE WAVE EQUATION
Define Pn G Q5(X,X n ) by Pn{x,y} = {Pnx,Pn'y} orthogonal (in X) projection of X onto Xn.
241
and note that Pn is the
Let En be the identity map from Xn to X and note that assumption A of Section 5.4 is satisfied. Define S : T>(S) -* X by T>(S) = {{x,y} S{*,y} = { - y , 4 z }
\ x G £ ( A ) , y G V} and
for
{x,y}eV(S).
Observe that assumptions H I , H2, H3 remain valid, with the same constants, if we replace both H and V with Xn. Let the operator An G 93(Xn) be the operator associated with 5-restricted to Xn. Hence, An is defined by (5.26). Define Sn{x, y} = {-y, Anx} for {x, y} G X n . Let us show that B and C of Section 5.4, with S, Sn in place of A, An, hold. Theorem 4.4.1 implies that there exist c,M G (0, oo), which depend only on b, a,Mi,M2,M3, such that \\(Sn-X)-k\\<M(\X\-c)-k
for
|A| > c, A G R, n,ife>l.
Hence, the Hille-Yosida Theorem 4.3.5 implies that lle-^H < Mect
for
t > 0, n > 1.
This shows that assumption B of Section 5.4 is satisfied. Choose A < —c and {z, w} G X. Define {x^}
{Xn,Vn} = (Sn ~ X^Pn&M,
= (S ~
*)~1{*M>
In view of Theorem 4.4.l's proof note that —A2 < a + M^M^2 and that x = (A -f A 2 ) - 1 (w — Az), y = —Arc — z, and similarly for x n , yn. Hence \xn-x\
= <
\{An + \2)-1(P^w-XP'nz)-{A + 2 Un + X^Py-iA + X )-^] 2
l
X3)-1{w-Xz)\
2
+\MWn + X )~ Kz -{* + A )~^l +\X\\(An + X2rl(P^z-PZz)\.
(5.34) (5-35) (5.36)
Theorem 2.8.11 implies that expressions (5.34) and (5.35) converge to 0 as n -» oo. Lemma 2.8.10 implies that for some c\ < oo, depending only on A, a, Mi, M2, M3, we have
IM„ + AV(f*;*-iMi < ctWKz-pZzw < < <
CI(||P^-Z||
+ ||Z-PM|) 2c1\\P^z-z\\ 2c1M1|P^-z|
242
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
which, by H4, converges to 0 as n —> oo. Thus, expression (5.36) and hence \xn — x\ converge to 0 as n —> oo. Since \\yn - y\\ = \\X(x -xn)
+ z - P'nz\\ < Mx\\\\x
- xn\ + Mx\z - P'nz\,
we have that \\yn — y\\ also converges to 0 as n —> oo and therefore assumption C of Section 5.4 is satisfied. Define H{t,x) = {0,/(t,x)} for 0 < t < r, x G X and note that D of Section 5.4 holds. Let {w, v} be the mild solution of (5.33) as given by Theo rem 5.1.2, i.e. {u,v} G C([0,r],X) is such that {u(t),v(t)}
- {u0,v0} + S [ {u,v} = [ { 0 , / } . Jo Jo
Hence, u G C([0,r],V) and u(t) — UQ — / v(s)ds = 0 Jo v(t)—vo + A
u(s)ds = / f(s,u(s),v(s))ds. Jo Jo Therefore u(0) = u0, u e Cl{[^,r\,U), v! = u, u'(0) = v0 and u'(t)-v0
+A
Jo
u{s)ds=
/ / ( s , u(s),u(s))ds Jo
for
t G [0,T].
If ze V, then, f o r t e [0,r], (u'{t),z) - {v0,z)+$(j
u(s)ds,zj
= f
(f{s,u{s),u'(s)),z)ds.
Theorem 2.8.12 gives the representation S(rc, y) = (BGx, Gy) for x, y G V; hence $( I u{s)ds,z\
= (BG I u(s)ds, Gz ) .
Since u G C([0,r],V), we have that Gu G C{[0,T],H) Theorem 4.2.10 implies G Ju = J Gu. Hence 5 ( I u(s)ds,z j = ( I BGu{s)ds,Gz\
by Theorem 2.8.12, and
= f
${u(s),z)ds
and therefore (uf(t),z)
- (v0,z) + / $(u{s),z)ds Jo
= / Jo
{f(s,u(s),u'(s)),z)ds,
5.8. FRIEDRICHS
EXTENSION
AND GALERKIN APPROXIMATIONS
243
implying that u has all of the claimed properties. G C 1 ([0,r],X n ) satisfy
Let {un,vn} {Un,Vny
+ Sn{un,Vn}
= PnH(t,
{Un,W n }),
{^n(0), Vn(0)} =
Pn{uQ,V0},
i.e. vn = u'n and < + A nu n
= P»f(t,un,u'n),
un(0) = P > 0 ,
<(0) = P > 0 ,
which is, in view of (5.26), equivalent to the formulation given in the Theorem. Theorem 5.4.7 implies that {un,vn} converge to {u,v} and this completes the proof. □
5.8
Friedrichs Extension and Galerkin Approximations
Theorem 2.12.6 showed that an elementary algebraic condition implies convergence of Galerkin approximations to the solution of an elliptic problem, where the elliptic operator is the Friedrichs extension determined by the basis functions. The same holds for parabolic and wave type equations. In the case of parabolic equations we have: Theorem 5.8.1 Suppose that (a) 7i is a complex Hilbert space (b) (j)n EH for n > I, Vn = s p a n { 0 i , . . . , cj)n} and U^=lVn is dense in % (c) S is a linear operator in % with domain T)(S) = Uj^Vn and such that there exist r G R and M G (0, oo) such that \Im(Sx,x)\
<MRe(Sx-rx,x)
for all
xeT>(S)
(d) r G (0, oo), H : [0, r] x % —> % is continuous and such that for some L < oo \\H(t,x)-H{t,y)\\
for
0 < t < r,
x,yeH
xeH.
Then for every n > 1 there exists a unique un G C 1 ([0, r], Vn) such that for all z G Vn we have (un(0),z) = (x,z) and (u'n(t),z) + (Sun(t),z)
= (H(t,un(t)),z)
for
t€[0,r].
Moreover, lim sup ||wn(£) — u(t)\\ = 0, ->°°o
u(0) = x and A is the
244
CHAPTER
5.
WEAKLY
NONLINEAR
EVOLUTION
EQUATIONS
PROOF T h e o r e m 2.12.1 implies t h a t all assumptions of Section 5.6 are satis fied. Hence, t h e assertions of t h e T h e o r e m follow from T h e o r e m 5.6.1. □ EXAMPLE 5.8.2 To see how easy it is to apply the above Theorem to even abstruse problems, consider the following PDE: ut = (poux)x + piux + P2U + p 3 sin(Reu) u{x, 0) = g(x)
for
for
x G E, t > 0
i6l,
where (1) Po,Pi,P2,P3 are complex valued functions in L°°(E) (2) p 0 G C^R)
and p'0 G L°°(E)
(3) there exists c G (0, oo) such that \pi(x)\2 + |Impo(x)| < c 2 Rep 0 (z)
for
£ GE
(4) g G L 2 (E) is complex valued. Note that po can vanish on an interval. Let H be the complex L 2 (E) and let <j>n{x) = xn-1e~x2/2
for
n > 1, x € E.
Define Vn = s p a n j ^ , . . . ,<£n} and T>(S) = U™=1Vn. Note that v G D(5) iff v(x) = P(x)e~x I2 where P is a polynomial. Theorem 2.9.8 implies that T>(S) is dense in %. Define Sv = -(pov'Y - piv' - p2v for v G D(5). If D G X>(S), then cx>
/
Po|v'| 2 - p i v ' t J - p 2 | u | 2 . -oo
Hence Re (Sv,v) >w2-
cw\\v\\2 - HP2IUMI2
|Im ( 5 V , V ) | < c2w2 +cw\\v\\2 + ||p2||oo|klll, where w2 = J |u'| 2 Repo, w > 0. Thus, for any M > c2 one can find r so that (c) is true and hence Theorem 5.8.1 applies.
5.9.
EXERCISES
5.9
245
Exercises
1. Show that if X is a Hilbert space and Rr is the retraction map given in Theo rem 5.1.4, then \\Rr{x) - Rr(y)\\ < \\x - y\\ for all x,y G X, r > 0. 2. Give the sharp upper bound for the blow up time of the solution of ut(x,t)
+ eu(x'*}
= A£u{x,t)
for xeRn,
t>0
for x e Mn,
u(x,0) = u0(x)
where UQ G C(E n ,E) is such that there exists lim^^ocUofa)
E R.
3. Give the sharp upper bound for the blow up time of the solution of ut(x, t) = Aeu(x, t) + u(x, t)2
for
a;Gln,f>0
for x e Mn,
u(x,0) = u0(x)
where UQ E C(R n ,M) is such that there exists limja-i^oo u0(x) > 0. 4. Prove that eA(t has the same integral representation as Q in Example 4.1.5. 5. Suppose that Aci -f Ci+i 4- c%-\ = 6di Show that ci = Y^=i Gn{hj)dj GB(ili) = G
for 1 < i < n, CQ = c^+i = 0.
for 1 < i < n, where
^ 0 = 6 t f f i ( ^ ^
for
l<j
/ ( i ) = X1 — \~l and A = 2 + \ / 3 . Show that this implies
Y,\Gn(i,j)\=3-3
\
j^l
^
A 2 -f A
for l < t < n .
2
Hence max Ki
|CJ|
< 3 max \di\. Ki
6. Show that the solution of the following system of ODEs, 4c- + c- +1 + c-_! + 2a - Ci+i - c»_i = 0 for 1 < t < n, c0 = Cn+i = 0, is Ci(t) = Y, -^~ ,=i
n +
l
\^2e~U}kt \k=i
sin
( * M s m O ' M J Ci (0) /
for 1 < i < n, t > 0,
246
CHAPTER 5. WEAKLY NONLINEAR
EVOLUTION
EQUATIONS
where 1 — cos ka 2 + cos ka
IT
n+ 1
Show that sup
\a(t)\ < 1.09677269 max |c*(0)|.
l0
l
7. Derive and prove convergence of the finite element approximation for d2u
du
1 du
..
x
i*(l,«) = ^ ( 0 , t ) = 0 u(x,0) = u0(x), where 0 < a ; < l , t > 0 and / , UQ are continuous. Let H consist of all measurable functions v for which
I
l
x|v(x)| dx < oo
o
and let the approximations be linear on intervals [ ^ , ^ ] , 1 < i < n, n > 2. 8. Using basis functions cos(n — l)nx, n > 1, prove convergence of Galerkin ap proximations for du _ d2u 1 du f(X t} ~di = dx^ + xdx~ + '
£<*«-!<'.<>u(x,0) = uo(z), where 0 < £ < 1 , £ > 0 and / , ^o are continuous. 9. Using basis functions xn(l — x), n > 1, prove convergence of Galerkin approxi mations for du d2u r, |X u(0,t) = w ( l , t ) = 0 u(rc,0) = tAo(^)? where 0 < a ; < l , £ > 0 and / , UQ are continuous. 10. State and prove the analog of Theorem 5.8.1 for the wave equation.
Chapter 6
Semilinear Parabolic Equations In this chapter we will extend the results of the preceding chapter to allow for less restrictive nonlinearities. While the exact definition of semilinear parabolic equations is left for Section 6.4, its typical representative is ut = Au + /(w, Vtx). The dominant term in the equation is the Laplacian, so, we need to control the nonlinearity in terms of the Laplacian which leads one to study its fractional powers. An equivalent approach is to define the nonlinearity in an interpolation space - a space between the domain of the Laplacian and the basic space.
6.1
Fractional Powers of Operators
Definition 6.1.1 Suppose that —A is the generator of a strongly continuous semi group {Q(t)}t>o on a Banach space X and that \\Q(t)\\ < Me~at
for
t>0
a
where a > 0 and M < oo. For such A define A G *B(X), for a G (—oo, 0), by Aax = —
1
r /
f°°
ra-lQ(t)xdt
for
x € X.
r(-«) Jo Let ®(X) denote the set of all such operators A in X. Observe that when a is a negative integer Theorem 4.3.2 implies that the above definition of Aa agrees with its usual definition. EXAMPLE 6.1.2 Let {^>i, y?2,. •.} be a complete orthonormal set in a Hilbert space 7i, let Ai, A2,... be complex numbers such that inf* Re A* > 0 and let a linear operator A in % be defined by 00
Ax = ^\k(z,
00
for xeQ(A)
k=i
=
{x€H\Yi\xk(x,
247
CHAPTER 6. SEMILINEAR
248
PARABOLIC
EQUATIONS
In view of Example 4.1.6 we have that A E &(X) and 1 r°° a 1 Xkt (Aax,ipk) = ——r t- - e- (x,
= \Ux^k)
for k > 1.
Hence, oo
A a x = ^A?(x,v?jfc)v?A;
for a < 0 ,
x6H.
The formula in the following Theorem can actually be used to define fractional powers of operators for operators in a bit larger class than <3(X). Theorem 6.1.3 If A e &{X), then POO
A_a =
sin7ra / 7T
PROOF
x-a^A
+ xyidX
for
a€
(0,i).
Jo
Theorem 4.3.2 implies that for all x G X roo
(A + \)~lx
=
roo
/ Jo
/ Jo roo
A - ^ A + AJ-^dA
e-XtQ{t)xdt
(A>0)
roo
\-ae-XtQ{t)xdtd\
= Jo
Jo
roo
= =
/ ta-lT(l-a)Q(t)xdt 10 Jo T(a)T(l - a)A~ax. D
Lemma 6.1.4 If A e <5(A"), then AaA^ = Aa+P for a, (3 < 0. PROOF
For all x e X we have
r°° ( - « ) Jo i
r
~x
i
roo roo
poo roo
i
roo
roo
r(
r V ^ W o /„ *-a-1C-0-"-lO(r)*** 1
z* 00
6.1.
FRACTIONAL
POWERS OF
249
OPERATORS
Lemma 6.1.5 If A e ®(X), then, for all a < 0, Aa is one-to-one and the range of Aa is dense in X. PROOF Pick -n < a. Since A~n = AaA-n-a we have that Jl(A~n) C R(Aa). Theorem 4.3.1 implies that $.(A~n) = T>(An) is dense in X and so is R(Aa). UAax = 0, then A~nx = A~n-aAax and therefore x = 0.
= 0, hence A~n+lx
= AA~nx
= 0,..., D
This makes possible the following: Definition 6.1.6 For A G <5(X) and a G (0,oo) let Aa be the inverse map of A~a, i.e. T>(Aa) = R(A-a), AaA~ax = x for x e X. Define A0 to be the identity map on X. EXAMPLE 6.1.7 When A is defined in a Hilbert space %, as in Example 6.1.2, then it is easy to see that for every a > 0 oo
oo
Aax = Y;Xk(x,
= {xeH\Y,\Xk(x>
k=i
k=i
Theorem 6.1.8 If A £ <&(X), then (1) Aa is closed, densely defined, 0 G p{Aa), ( A a ) _ 1 = A~a for all a > 0 (2) AaQ(t)x
= Q{t)Aax foraeR,t>0,xG
(3) D ( A a ) c D ( # ) a
T>(Aa)
ifa>/3
a
(4) A +Px = A A?x for a,/3eRandx<E
V{A^) n V{Aa+P)
(5) WA^xW < 2(Af + l)||Aa;|| a ||a;|| 1 - a for all a G [0,1] and x G T>{A) (6) for each A G R there exists c < oo suc/i £/ia£ ||x - eAtQ(*)rr|| < c(l + ext)ta\\Aax\\ PROOF
for
a G [0,1], x G T>(Aa), t > 0.
Definition 6.1.6 and Lemma 6.1.5 imply (1).
Definition 6.1.1 and Theorem 4.2.8 imply (2) when a < 0 and for a > 0 it follows from the fact that ( A " ) - 1 = A'a. If a > p > 0, then, by Lemma 6.1.4, A~a = A~^A^~a When a < 0, /3 < 0, then Lemma 6.1.4 implies (4).
which implies (3).
CHAPTER 6. SEMILINEAR
250
PARABOLIC
When a < 0, (3 > 0, a + p > 0, then A'^^A01 implies A~a~^AaA^x = x which implies (4).
EQUATIONS
= A~P applied to A$x
When a < 0, p > 0, a + f3 < 0, then Aa+0A-^{A^x)
= Aa{Al3x)
implies
(4). When a > 0, 0 < 0, a + 0 > 0, then A^A~a-^ implies -A^x = A~aAa+f3x which implies (4). When a > 0, /3 < 0, a + /? < 0, then A~aAa+0x implies (4). When a > 0, /3 > 0, then A~0A~a A-PA-aAa+(3x = x which implies (4).
= A~a applied to ,4 a + / 3 x = ^ x for x G X which
= A " a _ ^ applied to A a + / 3 x implies
To prove (5) assume that x E T)(A), a G (0,1). Theorem 6.1.3 implies
\\Aax\\ = P^Axll < S m 7 r ( 1 ~ a ) [°°X^UA + X^AxWdX. JO
K
Since \\(A + A)"1!! < M/A, \\A(A + A)"1!! < M + 1 for A > 0 we have Pa*lh 77 7 < sm7r(l-a)
eacTl(M
+ l)||x|| + ^ ( l -
< ea{M +
l)\\x\\+e?-lM\\Ax\\
< \\Aax\\
r A Q - 1 ( M + l)||x||dA+ r°A°- 2 M||Ai;||dA J0 Je ot)-lM\\Ax\\
for all e > 0. Minimizing this expression, see (3.15), gives the bound in (5). To prove (6) let z = x - extQ{t)x.
(2), (4) and (5) imply
||;z|| = p 1 - 0 ^ * - 1 * ! ! <2(M + Hence, using \\Aaz\\ = \\Aax - extQ(t)Aax\\
implies (6).
l)\\Aaz\\1-a\\Aa-lz\\a. < M ( l + e At )||A Q x|| and
Aa~lz
=
(A-X)A-1
ll^a"^l|
<
ci / eXs\\Aax\\ds
f Jo
eXsQ(s)Aaxds +
ext)t\\Aax\\ D
The following Theorem is the key tool for obtaining bounds in terms of fractional powers of operators - which are used to control the nonlinear term in semilinear parabolic equations.
6.1. FRACTIONAL
POWERS OF
OPERATORS
251
Theorem 6.1.9 If Ae ®(X) and B is a closed linear operator from X into another Banach space Y such that T>(B) D T)(A) and that for some /3 G [0,1), c < oo we have \\Bx\\Y < c\\Axf
\\x\\l"fi
for
xeT>(A),
(6.1)
then T)(B) D T)(Aa) and BA~a € (3. PROOF
Choose 7 € (/?, 1), 7 < a and x e £>(A7). Theorem 6.1.3 implies
x = A-iA^x
= f°° f(X)d\
where
/(A) = ^^^l\-r(A
+
\)~lA^x.
(6.1) implies that Bf E C((0,oo), Y) and
l|B/(A)||y < cllA-^A + AJ-^^II^IIA-^A + A)- 1 ^^!! 1 -^ < c\-*\\A(A + ArYiKA + A)- 1 !! 1 -^!^!!. Since ||(^4 -f- A)"1!) < M / ( A + a), ||A(A +A)" 1 !! < M + 1 for A > 0, we have ||J5/(A)|| y
<
c(M + l)A- 7 (A + a ) ^ 1 | | ^ x | |
00
L
\\Bf(x)\\Yd\ <
aWxl
Hence Bf is Bochner integrable on (0,00) and therefore Theorem 4.2.10 implies that x = ff e V{B) and \\Bx\\y < ci||A 7 a;||. Clearly, V(B) D T)(Aa) and ||BA- a || = | | 5 A - ^ ^ t t | | < C l | | ^ - t t | | . D
EXAMPLE 6.1.10 Suppose p G (n/2,oo), p > 1 and let I be the identity map from Lp(Rn) into Cu(R n ). Note that I is closed. Using (3.14) and the definition of A in Example 4.3.10 implies that
HJulloo^cllStillflHlJ"* for ueV(K). Choose a > 0. Note that ||e 5 t || < 1 for t > 0, ||A(a - A)"1!! < 2 hence ||/u|| 0 o<2c||(o-S)u||?||u||J"*
for
uGD(A).
Theorem 6.1.9 implies that, for every a > n/(2p), D((a - A) a ) C C B (K n ) and there exists C\ < 00 such that H/ulloo^cilKa-^ullp
for u <E D((a - A)«).
CHAPTER 6. SEMILINEAR PARABOLIC EQUATIONS
252
EXAMPLE 6.1.11 Suppose 1 < q < p < oo, /? = \ + § (± - ±) < 1 and 1 < k < n. Let V(B) consists of all u G Lq(Rn) such that Dku exists and Dku G L p (E n ). Define £ : V{B) C L*(Rn) ->• Lp(Rn) by £u = D*u for u_e 0 ( £ ) . Theorem 3.4.3 implies that B is closed. Using (3.16) and the definition of A in Example 4.3.10 implies that
||Bt*||p < c\\Kufq\\u\\\-e
for
ueV(K).
Theorem 6.1.9 implies that, for every a > 0,a > 0, D((a - A) a ) C V(B) and there exists c\ < oo such that ||Bti|| p
for u E T>((a-A)Q).
are such that T>(A) = D(B) and
\\(A-B)x\\
for
x€V(A)
(6.2)
for some a G [0,1), c < oo. Then V(A^) = £>(£^) for all 0 G [0,1]. PROOF
Assume /? G (0,1), x e X. Theorem 6.1.3 implies that B-'x
- A-?x = ™ ^
r
7T
JO
where/(A) = \-(3{B+\)-1(A-B)(A+\)-lx. 1
IKA + A)- !! < M / ( A + a),
f(X)d\
(6.3)
Note that for some a > 0,M < oo 1
||(B + A ) - | | < M / ( A + a)
for
A > 0.
Hence (5) of Theorem 6.1.8 implies ||^(^
+ A)"1!! < ci(A + a ) ^ " 1
for
A>0
and (6.2) implies MA-BHA
+ X^WKcifr
+ a)01-1
for
A>0
for some cx < oo. Thus \\B^f{\)\\ < c\A"^(A + a)a^-2\\x\\. Hence B^f is Bochner integrable on (0,oo) and Theorem 4.2.10 implies that / / G T>(B0). (6.3) implies A^x G 7)(B^) and therefore D(A^) C T>(B0). Since we also have (Chapter 1, Exercise 15) \\(A-B)x\\
for
x G D(£)
the above conclusion implies Ti(B^) C D ^ ) . Corollary 6.1.13 If A e <&(X), b>0 andae
[0,1], then T){Aa) = T>((A + b)a).
D
6.1. FRACTIONAL
POWERS OF
OPERATORS
253
One can actually show that (A + b)ax has a limit as b —>> — a (see Definition 6.1.1) for every x G D(A a ), a G (0,1). This enables one to extend the definition of Aa, a > 0 for the case when —A is the generator of a bounded strongly continuous semigroup. A Theorem similar to Theorem 6.1.9 can be proved in this case. Corollary 6.1.14 Under the assumptions of Theorem 2.8.6 we have T)({A-a)P)
= T>{{A*-aY)
= T>{{Ar-aY)
for
/?€[0,1].
PROOF Using the notation and assumptions as in the Sectorial Forms section, we have for x G T)(A), y G V ((A-Ar)x,y)
=
\({A-Ar)x,y)\
< b\x\\\y\\
\\(A-Ar)x\\
< b\x\
\\(A-Ar)x\\2 \\(A*-Ar)x\\
{d{x,y)-5(y,x))/2
<
62(5r(x,x)-a||x||2)/M3
=
b2((Ar-a)x,x)/M3
= \\(A-Ar)x\\
bM-l/2\\(Ar-a)x\\^\\x\\1/2.
<
Hence Theorem 6.1.12 implies the conclusion.
□
T h e o r e m 6.1.15 Suppose A is a sectorial operator and that a < b < Re A for all A G o~(A). Then, for every n > 1 there exists c < oo such that \\{A - a)ae~At\\ < ct~ae-bt
for all t > 0, a G [0,n].
PROOF Lemma 4.5.5 and Theorem 4.5.6 imply that A G 21(6 + 8, M, 6, X) for some 6 > 0, M and 6. Theorem 4.5.10 implies H e ^ H < cie'^5^ and Theorem 4.5.17 implies \\(A-a)e-At\\
+ t-l)e-(b+Vt
for
t > 0.
®(X). Theorem 4.5.14 and (5) of Theorem 6.1.8 imply
||(A - a)ae~At\\ < cAt-ae~bt
for all
t > 0, a G [0,1],
a
where c4 = 2(ci + l)cf c\~ . If 0 < a < n, then for all* > 0
(A-a)ae-At
=
[{A-a)ie-Aiy
\\(A-a)ae-At\\
<
(c4(t/n)-^e-^)n
=
c%nat-ae-bt.
CHAPTER 6. SEMILINEAR PARABOLIC
254
EQUATIONS
To simplify notation the following definition will be used. Definition 6.1.16 Let A be a sectorial operator in a Banach space X and a < Re A for all A G a {A). For a > 0, define Xa = Tf{(A - a)a) and ||z|| Q = | | ( A - a ) a a ; | |
for
x G Xa.
(1) of Theorem 6.1.8 implies that 0 G o-((A — a ) a ) , hence, Xa is a Banach space with the norm || • || a . Corollary 6.1.13 implies that Xa does not depend on the particular choice of a. Different choices of a give equivalent norms || • ||Q on Xa (Chapter 1, Exercise 15). Lemma 6.1.17 If A is a sectorial operator and n > 1, then, there exists c < oo such that || c - A (*+ fc )s - e-Atx\\a < c{e- 0, t > 0, 6 G [0,1], a G [0,n], /? G [0,a + £], a; G X^. PROOF Choose a e l such that a < Re A for all A G cr(A). Define A\ =A — a and * = A<*(e-A(t+V _ e -^*)a;. Note that z = ( A ? + ' " V * ) (e-A>he-ah
-
l)A^5A{x.
Theorem 6.1.15 implies
Ml < c^-^e-^Ue-^e-^
- 1)A^*A{x\\
and hence the bound follows from (6) of Theorem 6.1.8.
6.2
□
The Inhomogeneous Problem - Part II
Let us assume that A is a sectorial operator in a Banach space X, T € (0, oo) and / : [0, r] —> X is Bochner integrable and bounded. The purpose of this section is to derive some basic properties of v(t) = f
Jo
e-A^'s^f(s)ds
which will then be needed to obtain regularity of mild solutions of semilineax parabolic equations. T h e o r e m 6.2.1 v G (J 1 " a ([0,T],X a )nC a ([0,T],X) for all a G (0,1)-
6.2. THE INHOMOGENEOUS
PROOF
PROBLEM - PART II
255
Suppose a < Re A for all A G a(A) and A\ = A - a. Note that rt
rt+h
v(t + h) - v(t) = / gi(s)ds+
/
g2(s)ds
for 0 < t < t + h < r, where 9l(s)
= (e-^-^
- e-^"s))/(5),
e~A^h-^f(s).
32(3) =
Note that rt+h-s
A<£9l(s) = -AAi1
A\+ae-Auf(s)du.
/ Jt-s
Hence the boundedness of / and Theorem 6.1.15 imply rt+h-s
\\A?gi{a)\\
< =
f \\A
/ u-l~adu Jt-s ci((*-5)-a-(t + ^-5)-Q)/a
Cl
t+h
/
hl-aCl/(a-a2)
<
rt+h
\\A<*g2{s)\\ds < c2
h-s)-ads
(t +
= hl-*c2l{l-a) and Theorem 4.2.10 implies that v G Cl-a([Q,T],Xa).
Since
\Ht+h)- v{t)\\ < ur'wHt+h) - v{t)\\^a the above result implies that v G C a ([0,r],X). Define
G(t)=
Jo
fte-A^a)(f(s)-f(t))ds
and note that v(t)
= G(t) + / e~Asf{t)ds /o
for
(6.4)
t G [0,r].
Lemma 6.2.2 / / i / G (0,1), a G [0,i/) and f G C " ( [ 0 , T ] , X ) , tfien, G(t) G X 1 + a /or t G [0,T] and AG G C ^ - a ( [ 0 , r ] , X a ) . PROOF
Choose 0 < £ < £ + h < r and note rt
G(t + h)-G(t)=
/ gi(s)ds+ 7o
rt
rt+h
/ p 2 (s)ds+ / ^o ./t
9s(s)ds
(6.5)
CHAPTER 6. SEMILINEAR
256
PARABOLIC
EQUATIONS
where 9l(s)
= (c-**+ h -> - e - ^ ' - s ) ) ( / ( 5 ) - /(*)) 92(s) = e-^t+h-s\f(t)-f(t
+ h))
A t+h s
g3(s) = e- ( - \f(s)-f(t
+ h)).
Since A^e-A(t+h-s)
_ e-A(t-s)j
=
e-A(t-s)/2A(e-A{h+(t-s)/2)
_
e-A(t-s)/2j
Theorem 6.1.15 and (4.39) imply that \\Agi(s)\\a < c2h(t - s + 2h)~l{t -
a)"-"'1.
Hence rt
rt/h
/ \\Ag1(s)\\ads
{2 + r)-lru-a-ldr
(6.6)
Jo
(3) of Theorem 4.3.1 implies A' / '/■ ■ g2(s)ds = e~An(l - e-At)(f(t) Jo Jo Hence
IIII
cfll
II
\\A I g2{s)ds II Jo Theorem 6.1.15 and (4.37) imply t+h rt+h
- f(t + h)).
< c4hu~a.
(6.7)
rt+h
/
\\Ag3(s)\\ads X ^
a
a
a
then
25 differentiable, v' G C ^ - ( [ £ , r ] , X ) , w' G C([0,r],X)
v G C Q O ^ X 1 ) andAv G C^([e,r],X)
(^j V' + J4I; = / on [0,T].
6.3. GLOBAL
PROOF
VERSION
257
(6.4) and Lemma 6.2.2 imply that v(t) G T)(A) and Av(t)=AG(t)
+ f{t)-e-Atf{t)
for
*G[0,T].
(6.8)
Thus, Av G C ( [ 0 , T ] , X ) . Theorems 4.7.3 and 4.2.10 imply that v(t) + / Av(s)ds = [ Jo Jo
f(s)ds.
Hence v' + Av = f on [0,r] and t/ G C([0,r],X). (4.38) implies that e~Af Cu{[e,r],X) which implies that Av G CU([£,T],X). (3) and (6.8) imply v'(t) + AG(t) =
G
e-Atf(t).
Hence Lemmas 6.2.2, 6.1.15 and 6.1.17 imply that v' G C"-a{[e,r],Xa). and Theorem 4.2.10 imply that v : (0,r) -> Xa is differentiable.
This □
EXAMPLE 6.2.4 Consider an abstract delay equation of the form u'{t) + Au(t) = F(u(t - 1)) for t > 0,
(6.9)
u(t) = X is locally Lipschitz for some a G [0,1), a. Define v0(t) = n G C([0,l],X a ) we can define vn+1eC([0MXa)by vn+i(t) = e-^Vnfl) + f e-A^-s^F(vn(s))ds
for t G [0,1], n > 0.
Since v n (l) = vn+i(0) we can define u G C([— l,oo),X a ) by w(t) = vn(t - n + 1) for t € [n - 1, n]. It can be easily seen that u(t) = e- A V(0) + / e-A{t-s)F(u(s
- l))ds
for t > 0
and hence Lemma 6.1.17 and Theorem 6.2.1 imply that u G Cj/([-l,oo),X a ). Theo rems 6.2.3 and 4.5.14 then imply (6.9).
6.3
Global Version
When the nonlinearity is globally Lipschitz one can easily obtain global existence, uniqueness and continuous dependence. This is done in Theorem 6.3.3. Using the retraction map one can extend these results to locally Lipschitz nonlinearities - which is done in the next section.
CHAPTER
258
6. SEMILINEAR
PARABOLIC
EQUATIONS
Lemma 6.3.1 Suppose £ E (0,oo), p > 1, h E 1^(0, £) and 1 < q < oo. Define (Kg)(x) = f
h(x-
s)g(s)ds
g E Lq{0,£), x E (0,£).
for
(6.10)
Jo Then, K E »(L«(0,£)), limn_>oo \\Kn\\l/n = 0 and also Km E where 1/m < 1 — 1/p.
^{Ll{^£),L°°{^£))
PROOF Let g = h = 0 in R\(0, £). Since h E L X (E) and Kg = h * g on (0, £), Young's Lemma 3.1.5 implies that K E 1. Define l/pn = 1 - n/m for 1 < n < m. We claim that hi E L ^ R ) for I < i < m. This is obvious if i = 1. If the claim is true for some i < m, then 1 4- 1/pi+i = l/pi + 1/pi and hence Lemma 3.1.5 implies that the claim is true for i + 1. Therefore hm E L°°(R), *Tm G © ( L ^ O ^ ^ C M ) ) and |(lT»0)(s)| < | | M o o f ./o
foWl*-
Hence, for i > 1, K ^ ^ ^ I ^ I ^ ^ V - s r
1
! ^ ) ^
for
i€(0,«)
and ||X i m || < WehmWio/il If £ > 0 and n = im + j , with 0 < j < m - 1, then e~n\\Kn\\ < {\\K\\ley\\s-mthm\U» 5
^ 0
n 1 n
and hence ||i C || / < e for large enough n.
D
Theorem 6.3.2 (Gronwall) Suppose £ E (0,oo) and / G Ll(0,£) f(x)
h(x-s)f(s)ds
for
is such that
x E (0,£) a.e.,
w/iere /i G 1^(0, £), p > 1, /i > 0, 5 6 L 9 (0,£), g G [l,oo]. £e* if G !8(L 9 (0,£)) be given by (6.10). Then, / < (1 - K)~lg = g + Kg + K2g + • • •. PROOF Note that / < g + Ify + • • - + # n p + i ^ n + 1 / for n > 0. Hence Lemma 6.3.1 and Theorem 1.6.8 imply the assertion. □ Theorem 6.3.3 Suppose A is a sectorial operator in a Banach space X, a E [0,1), r G (0,00), H : [0, r] x Xa —>> X is continuous and such that for some L < 00 HtfftaO-tffty)!! < £ | | z - y | U
/or 0 < * < r, z , j / G X Q .
6.4. MAIN
RESULTS
259
Then, for each UQ G Xa there exists a unique u G C([0,r],X a ) which satisfies u(t) = e-Atu0
+ / e-^-^His, Jo
u(s)) ds
for
0
r.
Moreover, if G : Xa -+ C([0,r],X Q ) is given by G(uo) = u, then there exists c < oo such that \\G(x)(t)-G(y)(t)\\a
0 < t < r, x,y G Xa.
for
(6.11)
PROOF Abbreviate Y = C([0,r],X Q ) and note that Y is a Banach space with the sup norm (Example 1.3.2). For u G Y define Tu by (Tu)(t) = e-Atu0
+ f e-A^-s^H(s,u(s)) Jo
ds
for
0 < t < r.
Continuity in Xa of t -> e~Atuo and Theorem 6.2.1 imply that Tu G Y for u G 7 . If u,v G y , then Theorems 4.2.10 and 6.1.15 imply ||(Tt*)(t) - (Tv)(t)\\a
- v(s)\\ads.
(6.12)
Hence \\Tju - T^v\\a < K^\\u - v\\a where K is given by (6.10), with h(s) = c i s ~ a , q = oo. Lemma 6.3.1 implies that \\Kn\\ < 0.5 for some n and hence \\(Tnu)(t) - (Tnv)(t)\\a
< 0.5 sup \\u(s) - v(s)\\a
for
0 < t < r.
0<S
Thus, the Contraction Mapping Theorem 1.1.3 implies existence of a unique u £Y such that Tu = u. Choose x, y G X Q and let u — G(x),v = G(y). For 0 < t < r we have ||u(t) - «(t)|| a < c 2 ||x - y|| a +
Cl
/ (t - s)-a\\u(s)
-
v(s)\\ads.
Jo
Hence the Gronwall Theorem 6.3.2 implies (6.11).
6.4
□
Main Results
In this section assume that (1) A is a sectorial operator in a Banach Space X and a < Re A for all A G cr(A) (2) a G [0,1) and U is a nonempty open subset of Xa
CHAPTER 6. SEMILINEAR PARABOLIC
260
EQUATIONS
(3) T E (0, oo], F : [0, T) x U -> X is such that for every t E [0, T) and a; E ZY there exist T? E (0,1] and c, 8 E (0, oo) such that \\F(t2,X2) ~ F(tUX!)\\ < c(\t2 - hf + \\x2 - *i|U)
(6.13)
whenever U E [0,T), i j G W and |*i - £| + ||x» - z|| a < 5 for i = 1,2. For r E (0,T] let §(r) denote the collection of u 6 C ( [ 0 , T ) , W ) such that u'(t) exists, ix(t) E T)(A) and u'(t) + Ati(t) = F{t,u{t))
for all
t E (0,r).
(6.14)
We shall refer to such equations as semilinear parabolic equations. Observe that for u to belong to S(r), the set of solutions of (6.14), it is required that u be continuous in U and hence in Xa. In applications one can usually choose for a any number in a certain interval. It will be shown that the solution of an initial value problem does not depend on a particular choice of a. Note also that it is required that u(0) E U. Hence, solutions with 'rough' initial values are excluded though they may exist (Exercise 8). One can define S(r) in various other ways. However, in order to get uniqueness for a given initial value, some care is needed as Example 6.4.1 shows [17]. Several otherwise excellent texts make false uniqueness claims (Henry [11, page 54], Pazy [20, page 196]). EXAMPLE 6.4.1 Assume that a = 0, 0 < a < 1, U = Xa and that F(x) = \\Aax\\px. For p > 1 jot it is easy to find concrete examples such that JQ g{t)dt = oo where g(t) = \\Aae~Atz\\p for some z € X (Exercise 9). In such a case define w(0) = 0 and -I/P
u(t)= (l+p
I
g(s)ds)
Then, u ^ 0 and
(a)
ueC([o,i),x)nc1mi),x)
(b) u(t) e D(A) and u'(t) + Au(t) = F(u(t)) for 0 < t < 1 (c) F(u(-)) is Lipschitz continuous on (5,1) for every 6 E (0,1) (d) Sj ||F(u(t))||* < oo. Thus, this u and 0 are two different solutions of 6.14 with the same initial value, u does not belong to the solution set 8(1) because u is not continuous in U at t = 0. For a PDE version of this example see Exercise 10. EXAMPLE 6.4.2 Assuming only that / E C1 (Rn+1, E) and /(0) = 0 let us show that ut = An + /(u, Vu)
in Rn, t > 0
can be formulated as a semilinear parabolic equation in the complex X = Cto.
6.4. MAIN
RESULTS
261
Fix a G (1/2,1) and let Xa = D((l - A) a ). If u G X a , then, in view of Exam ple 6.1.11, D{U G X and we can define F(u) = f(Reu,ReVu)eX
ueXa.
for all
Choose any r G (0,oo). In view of Example 6.1.11, there exists c < oo such that if w G Xa, then ||w|| < c||iu||a and \\Diw\\ < c\\w\\a for all i = l , . . . , n . Let L be the bound of all first order partial derivatives of / while each of its n + 1 variables lies in the interval [—cr,cr]. Note that \\F(u) - F{v)\\ < (n + l)Lc\\u - v\\a
if ||u||« < r and ||v|| a < r.
Theorem 6.4.3 Suppose r G (0,T]. T/ien u G S(r) »/ and only if u G C([0,r),W) and u(t) = e" Ai u(0) + / e-A^-^F(s,u(s))ds
for all t G [0,r).
(6.15)
Moreover, if u G S ( T ) ; £/ien (a,) w',^4.14, F(-, w) G C# ( ( 0 , T ) , X ) . Moreover, if one can always choose •& > 1 — a in (6.13), then, u : (0,r) —> X^ is differentiable and u' G C#((0, r ) , ^ ) /or every 77 G [0,1 — a) (b) if ye
[a, 1] and u(0) G A"*, tfien u G C([0,r), X*).
PROOF If u G C([0,r),ZY) and /(*) = F(t,u(t)), then / G C([0,r),X). Thus, if u G S(T), then Theorem 4.7.1 implies (6.15). Assume now that u G C([0,r),ZY) and that (6.15) holds. Note that u(t) = e'Atu(0)
+g(t)
where
g(t) = [ e"A{^'8)f(s)ds. Jo
(6.16)
Choose any 0 < a < b < r. Since / G C ( [ 0 , T ) , X ) , Theorem 6.2.1 implies 0 6C ,1 - /J ([O,6],X /? )
for all
0 6(0,1).
(6.17)
Lemma 6.1.17 implies that for a < s < t < b and /3 G (0,1) we have \\e-Atu{0) - e-Asu{0)\\p
< c{t - s) 1 "^
which together with (6.16) and (6.17) implies that u G C 1 _ ^([a,6],X^). This and the assumed regularity of F imply that / is locally Holder continuous on [a, 6]. Hence, / G C"([a,b],X) for some v G (0,1). Actually, if one can always choose d > 1 — a in (6.13), then any v G (0,1) with v < 1 — a will do. Let v(t) = e-Atu(a)
+ f e-A{t~s)f(a Jo
+ s)ds
for
t G [0, b - a].
CHAPTER 6. SEMILINEAR
262
PARABOLIC
EQUATIONS
Theorem 6.2.3 implies that v'(t) exists, v(t) G 2)(A), v'{t) + Av(t) = f(a +1) for t G (0, b — a) and that ^ G q M - o ] , I ) , D ' e H M - f l U " )
for
£ G ( 0 , 6 - a ) , r/G[0,z/).
On the other hand it is easy to see, by rearranging (6.15), that v(t) = u(t + a) and since a, b are arbitrary we see that u G S(r) and (a) holds. If 7 G (a, 1), then (b) follows from (6.16) and (6.17) and the continuity of t -> e~Atu{0) in X*. If 7 = 1, then t -> e~Atu(0) is in ^ ^ ( [ O ^ ] , ! 0 ) by Lemma 6.1.17. Hence, (6.16) and (6.17) imply that u G CP([0,b],Xa) for some (3 G (0,1). Thus, / G C{[0,b],X) for some v G (0,1) and therefore An G C([0,6],X) by Theo rem 6.2.3. □ Now the results of Section 5.2 are going to be adapted to semilinear parabolic equations. Lemma 6.4.4 Suppose r G (0,T) and w G C([0,r],ZY). Then there exist S > 0, L < oo and a continuous H : [0, r] x Xa —> X such that (i) ifte
[0, r] and \\z - w(t)\\a < 8, then z eU and H(t, z) = F(t, z)
(ii) \\H(t,x)-H(t,y)\\ PROOF
< L | | z - y | | t t forte
[0,r], x,y G X " .
For £ G [0, r] choose J(t) > 0 and L(t) < oo such that x,yelA
and
\\F(s,x) - F{s,y)\\ < L(t)\\x - y\\a
when ||a;-iu(t)|| a < £(t), ||y-w(t)|| a < S(t) and | s - * | < 6(t). Let /i(t) G (0, <5(t)) be such that ||w(s) - w(£)||Q < < W / 2 i f I 5 - *l < A*W- Compactness of [0,r] implies that [0,r] C B(ti,/i(*i)) U ••• U5(f n ,/i(i n )) for some £i G [0,r]. Let L = 2 max; L(£;) and 5 = mini S(ti)/4. A short verification gives that if t G [0, r] a n d x , y G#(w(t),2<5), then x, y G ZY and ||F(*,x) - F ( * , y ) | | < L\\x - y\\a/2. Define #(*,x) = F(t, w(t) + Rd(x - w(t))) for t G [0,r] and a; G X Q , where i?<j is the retraction map in Xa (Theorem 5.1.4). □ Existence for the initial value problem: Theorem 6.4.5 For each u0 G U there exist 6 G (0,T) and u G S(0) such that ii(0) = UQ. Furthermore, limn->oosup 0 < t<e \\un(t)-u(t)\\a = 0 where uk G C([O,0),W) are defined by u\ (t) = UQ and uk+1{t) = e-Atu0
+ [ e-A^-s^F{s,uk{s))ds Jo
for
t G [0,6), k = 1,2,...
6.4. MAIN
RESULTS
263
PROOF Pick r G (0,T) and let w(t) = u0 for t G [0,r]. Let J and if be as given in Lemma 6.4.4 and let u G C([0,r],X a ) be as given by Theorem 6.3.3. Continuity of u implies that for some 0 G (0, r] we have for t G [0,0] that \\u(t) - w{t)\\a < 8/2. Hence H(t,u{t)) = F{t,u(t)) and therefore u G S(0) by Theorem 6.4.3. (6.12) implies that T is a contraction when r is small enough, which implies that \\uk — w\\a < 8 on [0,9] for k > 1 and that Uk -+ u. □ Uniqueness for the initial value problem: T h e o r e m 6.4.6 If 0 < n < 9 < T, we w(t) =u(t) forte [0,/i).
§(/i), u G §((9) and w(0) = u(0),
then,
PROOF Choose any r G (0, /JL) and let £ and if be as given in Lemma 6.4.4. Let r' be the largest number in [0, r] such that \\u(t) — w(t)\\a < S for t G [0, r ' ) . Note that r ' > 0. Note that H(t,u{t)) = F(t,u(t)) and H(t,w{t)) = F(t,w{t)) for £ G [0,r'). Hence, Theorems 6.4.3 and 6.3.3 imply that u = w on [0,r'). Continuity of w — u; and maximality of r' imply that T' = r. □ Solutions of real equations, with real initial conditions, are real: T h e o r e m 6.4.7 Suppose that Y is a closed real subspace of X such that (1) (A - X)~lY C Y for some (2) F(t,x)
\
G Y when t G [0,T) and x G Y flU.
Suppose also that u G §(0) for some 9 G (0,T] and that u(0) G Y. u(t)eY
for all
Then
te[O,0).
PROOF Choose any r G [0,9) such that u(t) G Y for £ G [0, r]. Theorems 6.4.5 and 6.4.3 imply that there exist 6 G (0,0 — T) and u G C([0, £),^0 such that v(t)
= e-Atu(r)
+ /" e-^-^Fir + s, v(s))ds for all t G [0, <J) io and that i; = linin-^ooVn for some vn G C([0,5),U) given by ^i(t) = u(r) and v n + i(t) = e~Atu{r) + f e~A^F{T + a,vn(a))
CHAPTER 6. SEMILINEAR
264
PARABOLIC
EQUATIONS
EXAMPLE 6.4.8 The solution of the generalized heat equation, given in Example 6.4.2, is real when the initial value is real (see also Example 4.6.3). Continuous dependence on the initial value: Theorem 6.4.9 Suppose 0 < r < /i < T and w G S(AO- Then there exist e > 0 and c < oo such that for every x G Xa, with \\x — w(0)|| Q < e, there exists u G S(r) such that u(0) = x and ||ix(t)-tw(t)|| a
for all t € [ 0 , r ) .
(6.18)
PROOF Let 8 and if be as in Lemma 6.4.4. Let c be as in (6.11) and let e G (0,8/c). Suppose \\x — w(0)\\a < e. Let u = G(x), where G is as in Theorem 6.3.3. (6.18) follows from (6.11) and the fact that w = G(w{0)) on [0, T]. Since ce < 8 Lemma 6.4.4 implies that H(t, u(t)) = F(t, u(t)) for t G [0, r] and hence Theorem 6.4.3 implies that u G S(T). □ Maximal interval of existence: Theorem 6.4.10 Suppose UQ G U. Then, there exist r G (0,T], u G §(r) such that u(0) = UQ and which also have the property that r > /d for all /i G (0,T] for which there exists v G §(/x) with v(0) = u$. Moreover, if r < T, then either sup 0 < £ < T \\F(t,u(t))\\ = oo or there exists x G U\U such that limt_>r \\u(t) — x\\a — 0. PROOF Theorems 6.4.5 and 6.4.6 imply the first assertion. Suppose r < T and sup0
= e-Atu(0)
+ [ e-^-^His.vis^ds Jo
for
t G [0,r']
and its uniqueness implies that v = u on [0,r). Continuity of v — u implies that for some fi G (T,T'] we have that \\v(t) — u(t)\\a < 8 for t G [0,/i] and hence H(t,v(t)) = F{t,v{t)) for t G [0,/x]. Therefore Theorem 6.4.3 implies that v G §(/i) - which is a contradiction to the maximality of r. □ Stability:
6.4. MAIN
RESULTS
265
T h e o r e m 6.4.11 If a > 0, T = oo and if for each e>0
there exists 8 > 0 such that
and \\F(t,x)\\ < e\\x\\a whenever x G Xa,
xeU
\\x\\a <8
then there exist \i > 0 and c < oo such that for every x G Xa, exists u G §(oo) such that u(0) = x and IK^)||a
for
andt>0,
with \\x\\a < fj,, there
t>0.
In view of Theorem 6.1.15 we can choose M, 0 G (0, oo) such that
||e-^|| < Me-(fl+^,
||(A - a)ae~At\\ < Mt^e^^1
Choose e > 0 so that eM6a~lY(l given. Choose /i G (0,8) so that
for
* > 0.
- a) < 1 and let the corresponding 8 > 0 be
2M/i < 6(1 - eM0a-lT(l
- a)).
(6.19)
a
Choose x G X with ||rc||a < fi and pick the maximal r > 0 such that there exists u G S(T) with u(0) = x (Theorem 6.4.10). Note that for some rj G (0, r] we have that ||u(£)|| a < 8 for 0 < t < rj. Let r' be the largest of such r\. Note that for a l H e [0,r') \\u(t)\\a
< Me-^+^WxWa
+ eM [^ (t -
5)-
a
e-(a+^-s)|K5)||Qd5
Jo
eat\\u(t)\\a
(t-s)-ae-^-^eas\\u(s)\\ads
< M\\x\\a + eM [ Jo
<
M\\x\\a + eM0a-lr(l-a)
sup e a s ||u( 5 )|| Q . 0<s
Hence sup en\u(s)\\a
o<s
Wt)l|a
<
y a
^
*
'" -l-eM0°-iT(l-a)
M|| g || t t e- f l *
"
l-EM0«-ir(l-a)'
This, the fact that \\x\\a < \x and (6.19) imply that ||u(t)|| a < V 2 for * € [0,r'). Continuity of w and the definition of r' imply that r' = r. This implies that ||F(t,ix(t))|| < e8/2 for t G [0,r) and that u stays away from the boundary of U. Thus, r = oo by Theorem 6.4.10. □ If the spectrum of A contains a point with a negative real part (the point does not have to be isolated) and if F is genuinely nonlinear at 0, then we have instability:
CHAPTER 6. SEMILINEAR
266
PARABOLIC
EQUATIONS
Theorem 6.4.12 Suppose that Rez < 0 for some z G o-(A), T = oo and that there exist 8,0,M G (0,oo) such that xeU
and \\F(t,x)\\ < M\\x\\}+e whenever x G Xa, \\x\\a <8 andt>0.
(6.20)
Then there exists c > 0 such that for each e > 0 there exists u G S(r) for which |K0)|| Q < e, PROOF
u(0) G n£° =1 2)(,4 n ).
||u(t)|| Q > c /or some t G (0,r),
Define ZQ = — i n f ^ ^ ) Re £ and choose A, fi so that 0 < fj, < ZQ < A
a n d fi(l + #) > A.
Note that Theorem 6.1.15 implies that || (A - a) Q e~ A t || < Mi t~aext
for all
t > 0.
(6.21)
Since there exists £ G cr(A) such that Re £ < —/i, we have, by Corollary 4.3.3, that sup t > 0 ||e _v4< xi||e _/i< = oo for some x\ G X. Define x2 = (A- a)-ae-A1xu
x = \\x2\\-1x2
a(t) = \\e-Atx\\ae-^,
Gn~ ^(A"),
b(t) = sup a(s). 0<s
Note that a is continuous, supt>0a(t)
= oo, a(0) — 1, ||e - A t a;|| a = a(£)eM<.
Pick c G (0, <J/3) so that M 2 = T(l - a)(/i(l + 0) - A) a ~ 1 3MMi(3c)^ < 1.
(6.22)
Let e > 0 be given. Choose to > 0 such that £a(to)eflto > 2c. Find the smallest t\ > 0 such that a(ti) = l + 6(t 0 ). Note that ti > t 0 ,
a(t) < a(ti) for t G [0,ti),
a(*i) = o(ti).
Choose €o so that eoa(^i)e /itl = 2c and note that eo < e, so <2c < 28/3. Theorem 6.4.10 gives u G S(r) such that it(0) = e^x and r is maximal. Note that ||w(0)|| a = so < e. Continuity implies that there exists T\ G (0,r] such that II^WIIQ < 1.5e0fe(*)e/it
for
t G [0,TI).
Let T2 be the maximal of such r\. Note that ||ti(*)||o < l.beob^e^
< l.beobfa)^*1
= 3c < 8
(6.23)
6.5. EXAMPLE: NAVIER-STOKES
EQUATIONS
267
for t e [0,r 2 ) D [0,*i]. Hence, (6.20) a n d (6.21) imply II / e-A^-^F(s,u(s))dS\\ \\J0
< MX M \\a
8)-aex^-^\\U{s)\\^ed8
f \ t Jo
s)~aex{
<
MXM
f (tJo
<
MiM(1.5£ O 6W) 1+0 [
(t-s)-aex^-^e^l+e">sds
Jo <
M i M ( 1 . 5 £ 0 & W e ^ ) 1 + ^ r ( l - a ) ( / i ( l + 6) -
\)a~l
<
0 . 5 e 0 W e ^ 3 M i M ( 3 c ) ^ r ( l - a ) ( / i ( l + 0) - A ) " " 1
-
e0b{t)e<xtM2/2.
(6.24)
Therefore (6.15) implies t h a t ||ti(*)||a < (1 + M2/2)e0b(t)e^
for
t G [0,r 2 ) n [0,*i].
(6.25)
If r 2 < t\, t h e n t h e continuity, t h e fact t h a t M 2 < 1 a n d t h e definition of r 2 would imply t h a t r 2 = r . However, in this case (6.23) a n d (6.20) would imply b o t h boundedness of F on [0, r ) a n d t h a t u stays away from t h e b o u n d a r y of U - which is not possible according to T h e o r e m 6.4.10. Therefore r 2 > t\. By construction, | | e - > U l u ( 0 ) | | a = 2c. Hence (6.15) a n d (6.24) imply
2c < ||ti(ti)|| a + 0.5£ 0 &(*i)e^ = IK*i)|| 0 + c a n d ||t*(*i)|| tt > c.
D
EXAMPLE 6.4.13 Since —A^ is a sectorial operator (Example 4.5.4) in the complex C^, we see that the nonlinear heat equation (Section 5.3) is a semilinear parabolic equation (a = 0) provided that we define F(t,u) — f(Keu) and modify U in the obvious way. Since the solution u € S m M , given by (5.10), is also the mild solution, we have, by Theorem 6.4.3, that u € S(r) and hence u'(t) = A£u(t) + f(u{t))
for
t e (0, r ) .
If /(c) = 0, /'(c) > 0 and / ' is Holder continuous near c, then a(—Ai — /'(c)) = [—/'(c), oo). Hence Theorem 6.4.12 implies that the constant solution c is unstable.
6.5
Example: Navier-Stokes Equations
Velocity of a fluid u = (iti, w2, us) and a scalar pressure p satisfy ut + (tx • V)tx = - V p + Aw.
(6.26)
268
CHAPTER
6. SEMILINEAR
PARABOLIC
EQUATIONS
The incompressibility of the fluid is specified by V • u = 0.
(6.27)
These are the Navier-Stokes equations. We shall apply the results of Section 6.4 to the case of a flow that is periodic in space and has 0 mean velocity. The periodicity cell is taken to be Q = (—7r, 7r)3. Define H0 = {ue L2{Sl) | Jnu = 0},
H = H0xH0x
H0.
H is a Hilbert space with an inner product {u,v) = (27r)~3 / uyvl + U2V2 + W3V3
Jo,
where u = (^1,^2^3) G H and v = (vi, 1*2,^3) G H. For k G Z 3 and u e H define £k(u) = (2TT)- 3 /" u{x)e-ik'xdx
e C3.
Completeness of the Fourier series (Example 2.1.7) and Theorem 2.1.10 imply that for every u G H, u(x) = y£ek(u)eik* (6.28) where ^ denotes, throughout this section, the sum over all k e Z 3 \{0} with conver gence in H and not necessarily pointwise. Note also that (u,v) = ^2£k(u)
• ik(v) for u,veH
(6.29)
and that u is real valued a.e. iff ^ ( M ) = £-k(u) for all A; G Z 3 . A formal differentiation of (6.28) implies that in order for u to satisfy (6.27) we should have k-ik(u) = 0 for all kel?. (6.30) (6.30) makes sense, unlike (6.27), for every u G H. Hence, the incompressibility condition (6.27) will be replaced with (6.30). Let X consist of all u G H which satisfy (6.30). Note that X is a closed subspace of H and hence a Hilbert space. (6.30) suggests the following decomposition of a given v G H. For k G Z 3 \{0} define pk = —i£k(v) • k/k • k and note that h(v) - iPkk J. *,
\ik(v) - ipkk\2 + \pkk\2 = \£k(v)\2.
Lemma 2.1.4 implies that we can define u G H and p G Ho by «(*) = £ ( / * ( « ) - tpfcfcje**
and p(x) = ^ P f c C * * .
(6.31)
6.5. EXAMPLE: NAVIER-STOKES
EQUATIONS
269
Note that u satisfies (6.30) and that v = u + Vp where Vp = t £ A^e**"*. Define Q G 05(H) and II G
and
IIv = p.
Observe that Q is a projection whose range is X, ||Q|| = 1, Q(Vp) = 0 and that u,p are real valued if v is real valued. Using (6.31) and (6.29) it is also easy to see that Q is self-adjoint. A formal differentiation of (6.28) implies that =-Y,\k\2ek(u)e>kx
(Au)(x)
and that Au satisfies the incompressibility condition (6.30) when u satisfies it. This suggests that we define the linear operator A in X by
%k x
(Au){x) = £ \k\ £k(u)e -
for
) u G V(A).
( J
(6 32)
-
Using (6.29) it is easy to see that A is symmetric. Theorem 2.6.3 implies that A is self-adjoint. Theorem 2.6.6 implies that 0 we have ) = {u G X | £ \k\W\lk(u)\2 (A^u)(x)=^2\k\2^k(u)eik-X
< oo} (6.33)
for
ueD(#).
Let us now analyze the (u • V)iz term in (6.26). Note that ( ( ^ ■ V R ^
^ + ^ + ^ for n = 1,2,3 (6.34) ox i OX2 UX3 where u = (1*1,^2,^3) &nd v = (v\, i>2>^3)- As is usual in this section, we take derivatives by formal differentiation of (6.28) and then we require that the resulting series converges in H, i.e. 1
oxm Thus, in order that all dun/dxm exist in H we should have ^ l&|2Kfc(w)|2 < °°- Hence (6.33) implies that we should have u G T>(All2). (6.33) and (6.29) then imply
/n Y
n,m=l
l l ^ f = (2*)3||41/2uf
for
ueViA1'2).
(6.35)
CHAPTER
270
6. SEMILINEAR
PARABOLIC
EQUATIONS
(6.28) implies \v(x)\ < Y, \tk(v)e**\ and since Cp = ( £ \k\~A(3)1/2
= £
\k\-V\k\W\lk(v)\
< oo for 0 > 3/4 we have that
Kx)| 2
for
v G E(A*), 0 > 3/4, z G
ft.
(6.36)
(6.34), (6.35) and (6.36) imply that (v ■ V)u 6 L 2 (0) 3 and since f(v-V)u=fi
Y(v(x)
' Wk(u)eikxdx
= i(2ir)3 ^ ( € _ f c ( v ) ■fc)4b(u)= 0
we have that (v • V)u € H. We also have the bound \\(v^)u\\ 3/4.
(6.37)
Fix now any a € (3/4,1) and let Xa be
2
for
u£Xa
| | F ( t i ) - F ( t ; ) | | < C a ( H | a + |H|Q)||ti-t;||a
(6.38) for
u,veXa.
(6.39)
Applying Q to (6.26) yields the following semilinear parabolic equation in X, u' + Au = F(u).
(6.40)
Let the initial velocity UQ G XQ be given. Theorem 6.4-10 implies that (6.40) has a solution u G S(T), on some maximal interval [0, r ) , swc/i £/ia£ u(0) = ^o- T/ie solution is unique by Theorem 6.4-6. If pressure is given by p = -n((ti.v)ti), then this pair u,p satisfies the Navier-Stokes equations. Since a(A) C [l,oo) ; Theorem 6-4-11 implies that if ||ito||a is small enough, then the solution u exists for all time (r = ooj and that \\u(t)\\a —> 0 exponentially.
6.6. EXAMPLE: A STABILITY
PROBLEM
271
Ifuo is real valued, then it is easy to see that Theorem 6.4-7 applies and thus u(t) is real valued a.e. for all t G [0,r). Let us show now that when uo is real valued \\u(t)\\ < e-'IKII
for
t€[0,r)
(6.41)
independently of the size of the initial \\uo\\a (it is an open question whether or not T = oo also for large ||uo||cJ(6-41) is called the energy bound. The following procedure for obtaining it is called the energy method and it can be adapted to many other evolution equations. Note that — |M| 2 = 2Re(u',u)
= -2Re(Au,u)+2ReR
on
(0,r)
(6.42)
where R = (F(u),u) = —{Q({u • V)u), u). Since Q is a self-adjoint projection we have that R = -((u ■ V)u,u). Using (6.28) gives
R = -{2<*fi £ ( * • lk,-k(u))lk(u) • M^y. k,k'
This, (6.30) and the fact that u is real valued imply that ReR — 0 and hence (6.42) implies that — |M| 2 = -2Re(Au,u) on at (6.32) and (6.29) imply that (Au,u) > \\u\\2. Hence
(0,r).
^(e2*|Kt)||2)<0 which implies the energy bound (6.41).
6.6
Example: A Stability Problem
There are many journals devoted exclusively to studies of fluid flows. In their articles, the governing equations are usually derived from the Navier-Stokes equations and one topic frequently addressed is stability of a certain solution. In the preceding Section 6.5 it was shown that the zero solution of the Navier-Stokes equations is stable. Stability analysis of nontrivial flows is usually more demanding and in unbounded domains an extra complication occurs which will be highlighted next. Studying impulsive stretching of a surface in a viscous fluid leads to xx(x,t) +(p2(x,t)
= (px(x,t) / Jo
4>(s,t)ds
One solution of this equation is Crane's solution e~x. solution is stable for small perturbations.
for
t > 0, x > 0.
(6.43)
We want to show that this
CHAPTER 6. SEMILINEAR
272
PARABOLIC
EQUATIONS
If g{x,t) = (j)(x,t) — e x, then (6.43) becomes
gt-9xx-9x
+ e-x [gx + 2g + / g)=-g2
+ gx
g-
(6.44)
It is required that p(0, t) = 0 and that g(x, t) decays as x increases. To apply stability Theorem 6.4.11 or instability Theorem 6.4.12 one needs to have the sharp lower bound on the spectrum of the linear operator different from 0. The following argument suggests that this may not be the case for equation (6.44). The spectrum of the following operator in 7i = £ 2 (0, oo) Aou = — uxx — ux
for
ueV(A0)
= W2{0,00)0^(0,00)
(6.45)
consists of those z e C such that (Imz)2 < Rez (Exercise 12). So, for this operator Theorems 6.4.11 and 6.4.12 do not apply. The actual linear operator in (6.44) consists of AQ and of a lower order perturbation of A$. So it does not seem very likely that the introduction of lower order terms would move the spectrum to the right - which is needed to prove stability. In such situations specifying a different growth rate at infinity can sometimes move the spectrum. After some experimentation, we find that the following requirements will solve the problem g(x, t) = e~x/2u{x, t)
and u(-, t) e L 2 (0, oo).
(6.46)
2
Thus, (6.44) will in effect be studied in a weighted L . Using (6.46) in (6.44) gives ut + Au = f(u)
(6.47)
where Au = -uxx + e~xux + ( \ + \e~x )u + e~x'2 [ e'^u
(6.48)
Jo
f(u) = -e-x>2u2
+ (ux - ti/2) r e~s/2u.
(6.49)
Jo
Let V = WQHO, OO). Define $ : V x V -► C by
5-(v, w) = I
(v'^
+ v'we~x + vw{ \ 4- \e~x ) + we~^2
Observe that if v e V and g(x) = f£ e~s/2v(s)ds, Re$(v,v)
= Re [ Jo
f
s/2 e- v(s)ds
J dx.
then
(\v'\2 + v've^ + \v\2{ \ + \e~x ) +gg') dx /•OO
=
i|ff(oo)|2 + Re / Jo
(|t/| 2 + | V | 2 ( i + | e - ^ ) + «'%"*) dx
oo
/ >
ill»ll?,2-
(K| 2 +,\v\ 2 (\ + 2e-*))dx
6.7. EXAMPLE: A CLASSICAL
SOLUTION
273
Therefore, 5 is a sectorial form on V (M 3 = 1/4, Mi = 1, a = 0). Lemma 2.4.5 implies that the linear operator associated with J is the operator A, given by (6.48), with the dom'ain V(A) = W2{0,oo) D 1^(0, oo). Theorem 2.8.2 implies that A is a sectorial operator and that its spectrum is contained in the half-plane Rez > 1/4. Integration by parts implies that we can express 5(v, w) — 5(w, v) as /•OO /
/
px
f 2v'we~x - vwe~x + we~x/2
/
rx
e-s/2v(s)ds
- ve~x/2
\
I e~s/2w(s)ds
) dx
which implies that \$(v,w) -$(w,v)\
< 4||w|| 2 |M|i,2
for all
w,v
EV.
Therefore Theorems 2.8.6, 2.8.12 and Corollary 6.1.14 imply that V(All2) that for some c E (0, oo) we have c-l\\v\\lfi<\\v\\1/2
for
= V and
veV.
If v e V, then |v(x)| 2 = 2Re / t / v < 2 | M | 2 | | t / 2 < lkll?,2. Jo Hence ||v||oo < IMIi,2 which implies that the nonlinearity / : V -> W, given by (6.49), is well defined ll/Wlh < 3|M| l l 2 |M| 2 < 3c||t;||? /2 for v € V and that l l / M - / ( « ) | | 2 < 3 | | t i ; - t ; | | i l 2 ( | H l i , 2 + IH|i l 2)
for
w,v
eV,
implying that / is locally Lipschitz in V. Thus, (6.4V is a semilinear parabolic equation (X = H, a = 1/2, U = V) and therefore Theorem 6.4-U implies stability of the 0 solution of (6.4V and hence stability of Crane's solution.
6.7
Example: A Classical Solution
We shall now study, in more detail, the following initial value problem: ut(x, t) = uxx(x, t) + g{u(x, t))
for
ti(0,t)=ti(*,t) = 0 for u{x,0)=u0(x)
for
0 < x < £, 0 < t < r,
(6.50)
0
(6.51)
0<x<£,
while assuming that £ e (0,oo), u0 e W$(0,£) is real valued and g e
(6.52) X
C {R,R).
274
CHAPTER
6. SEMILINEAR
PARABOLIC
EQUATIONS
In X = L 2 (0,^) define the linear operator A by Av = -v"
for
uG!D(A) = ^ ( O ^ n W ^ O ^ -
Choose a G (1/4,1/2) and define U = T>(Aa) = Xa. C
In view of Exercise 7, there
r
exists c < oo such that |M|oo < IMU f° v €U. Define F(i') = g(Rev) for v £U and note that | | f (»2) - JF-Cvi)!! < ||V2 - wi||
mBx\g'(r)\ \r\<mc
whenever V{ E U and \\vi\\a < m. Note also that W$(Q,£) = V(Alf2) C U and that ||u||i/2 = H-A1/2^ = IKII (Exercise 7). Thus, all assumptions of Section 6.4 are satisfied. Hence Theorem 6.4.10 implies existence of the solution u E S(r) on some maximal interval [0, r ) . The solution is unique and real valued (Theorems 6.4.6, 6.4.7). u is actually the solution of the abstract problem u 4- Au = F(u),
U(0)=UQ.
It does satisfy the original PDE with a proper generalization of derivatives. In many applications we do not really need to elaborate further on this point. However, there are many important techniques that require more precise pointwise estimates of solu tions. Use of a maximum principle, described below, is an example of such a technique. Let us demonstrate, through this example, how to obtain pointwise regularity of the solution. Since Theorem 6.4.3 implies that u(t) G WQ(0,£), we may assume that u(-,t) is continuous on [0,^] for all t G [0,T).
Theorem 6.7.1 u G C([Q,£\ x [0,r),M) and l^ti Uxx G C{[0,£\ x (0,r),R). Moreover (6.50), (6.51) and (6.52) are satisfied pointwise. PROOF Choose 0 < x < £, 0 0. Theorem 6.4.3 implies that u G C([0,r),X 1 / 2 ). Hence u G C ( [ 0 , T ) , C B ( 0 , £ ) ) (Exercise 7) and therefore there exists 8 > 0 such that max\u(r,t)-u{r,s)\<e/2
if
s G [ 0 , r ) , \t - s\ < S.
This implies \u(x,t) -u(y,s)\
< <
\u(x,t) -u(y,t)\
+ \u{y,t)
-u{y,s)\
\u(x,t)-u(y,t)\+e/2.
Hence the continuity of u(-,t) implies that u G C([0,^] x [0,r),E).
6.7. EXAMPLE: A CLASSICAL
SOLUTION
275
Theorem 6.4.3 implies that u : (0, r) -> X*1 is differentiate for 77 E [0,1 - a ) . In particular, we may choose rj = 1/2. This implies (Exercise 7) that lim max
u{r,t + h)-u{r,t)
u
7
/i->0 0
,r
( , t) = 0
and hence ut exists on [0, £\ x (0, r) in the classical sense. Since we have that ut E ( ^ ( ( O J T ) , ^ 1 / 2 ) , we can use exactly the same argument used to show continuity of u to show that ut E C{[0,£] x (0,r),R). Observation Au = g(u) — ut E C([0,1] x [0, r ) , R) implies the rest.
□
One can learn a lot about second order PDEs by pursuing the consequences of a solution having a local maximum or a local minimum. There are many recipes about how to do this, all of which usually go under the name maximum principle. One version is represented by Theorem 6.7.2. Note that, in (2) of the Theorem, it is assumed that only the indicated partial derivatives exist at the specified points and that / is an arbitrary real valued function satisfying (5). If / r (rc,t,r, 0) is bounded from above for x E (0,^), t E (0,T) and r E (—e,0], then (5) is satisfied provided that /(M,0,0) >0. Theorem 6.7.2 Suppose (1) £,TG(0,oo), veC([0J]
x[0,T],M) 7 D : (0,i) x (0,T)-> [0, oo)
(2) vt(x,t)
+ f{x,t,v{x,t),vx(x,t))
>D{x,t)vxx(x,t)
for
0<x<£,0
(3) v(0,i) > 0 and v(£,t) > 0 for t E [0,T] (4) v(x,0) > 0 for xe [0,i] (5) for some e > 0, L E [0, oo) we have that f(x,t,r, xe (0,£) andte (0,T).
0) > Lr for r E (—e,0),
Then, v{x,t) > 0 for all x E [0,£\ and t E [0,T]. PROOF Let us consider the case L = 0 first. Assume that u(a;o,£o) some x 0 E [0,£] and t0 E [0,T]. Define M(t) =
min
u(x,s)
for
<
^ ^ or
* E [0,T].
Note that M is continuous, nonincreasing, M(0) > 0 and M(t0) < 0. Hence we can choose t\ E (0,T) so that Af(ti) E (-£,0). Pick now x 2 E (0,€) and *2 G (0,ti] so that v(x2,t2) = M(ti). Since v(-,t2) has a minimum at x2 we must have vxx{x2,t2) > 0. Hence vt{x2,t2) > f{x2,t2,v(x2,t2),0) > 0 which
CHAPTER 6. SEMILINEAR PARABOLIC
276
EQUATIONS
implies that v(x2,h) < v(x2,t2) for some t3 < t2. Therefore, v(x2,ts) which is a contradiction. This proves the Theorem when L = 0. In general, define w(x,t) = v(x,t)e~Lt wt(x,t)
> Dwxx(x,t)
< M{t\)
and note that
+ fi{x,t,w(x,t),wx{x,t))
for
0 < x < £, 0 < t < T
where /i(x,t,r,«) = (f(x,t,reL\seLt) LreLt)e~Lt. LT Since /i(:r,£,r,0) > 0 if r G (—ee~ ,0) this reduces the problem to the case L = 0. D The proof of the following Corollary shows that uniqueness of the classical solution of (6.50) - (6.52) follows immediately from the above Theorem 6.7.2. Corollary 6.7.3 IfTe wt(x,t)
(0,r) and w e C([0,€\ x [0,T],R) is such that
= wxx(x,t)
+g(w(x,t))
iw(0,t)=iu(£,t)=0 w(x, 0) = uo(x)
for
0 < x < £, 0 < t < T,
for
0
for
0 < x < I,
thenw = u on [0,£\ x [0,T]. PROOF If v = u — w and f(x,t,v) = g(w(x,t) + v) — g(w(x,t)), assumptions of Theorem 6.7.2 are satisfied. Hence u — w > 0.
then the
If v = w — u and f(x,t,v) = g(u(x,t) + v) — g(u(x,t)), then the assumptions of Theorem 6.7.2 are satisfied. Hence w — u > 0. □ An easy exercise, left to the reader, gives Corollary 6.7.4 If a € (-00,0], g(a) > 0 and u0 > a, then Corollary 6.7.5 If b e [0,oo), g(b) < 0 and u0 < b, then
6.8
u>a. u
Dynamical Systems
Let M be a complete nonempty metric space with metric d and suppose that a family of maps Rt G C(M, M), t > 0, satisfies (a) Ro is the identity map (b) Rt(Rs{x))
= Rt+S(x) for all x e M and t, s > 0
6.8. DYNAMICAL
SYSTEMS
277
(c) the map t -+ Rt(x) belongs to C([0, oo), M) for every x E M. Such families of maps are called dynamical s y s t e m s on M. A strongly continuous semigroup of linear operators on a Banach space is an example of a dynamical system. Here is a nonlinear example: EXAMPLE 6.8.1 Suppose that g E CX(R, R) is such that g(a) > 0 and g(b) < 0 for some -oo < a < 0 < b < oo. Let M be the set of all v E W}(0,1) such that a < v(x) < b for all x E [0,1]. M is a complete metric space with metric given by the WQ(0, 1) norm. Corollaries 6.7.4 and 6.7.5 and Theorems 6.4.10 and 6.7.1 imply that for each u0 E M there exists u E C([0, oo), W^O, 1)) such that ut(x,t) = uxx{x,t) + g(u{x,t)) u(0,t) =u(l,t) =0 w(x,0) = u0(x)
for 0 < x < 1, 0 < t < oo for
0
for 0 < x < 1.
Define Rt(u0) = u(-,t). Continuous dependence Theorem 6.4.9 implies that Rt E C(M,M) for t > 0. Uniqueness Theorem 6.4.6 implies (b) and hence Rt, t > 0, is a dynamical system on M. We shall now introduce some basic notions used to study dynamical systems. The orbit of x E M is defined by 7 (x)
= {Rt{x) | t > 0}.
x is said to be an equilibrium point if 7(0;) = {x}. Define o;(x), the o;-limit set for x E M, to be the set of those y E M for which there exist tn E (0,00), n > 1, such that lim tn = 00 and lim Ru (x) = y. n-+oo
n-+oo
A set i C M is said to be positively invariant if Rt(A) C A for £ > 0. A set K c M i s said to be invariant if for each x E K there exists u E (7(IR, X) such that u(0) = x
and
Rt(u(s)) = u(t + 5)
for
< > 0 , s G M.
Theorem 6.8.2 J/ the orbit of x E M is relatively compact, then UJ(X) is nonempty, compact, connected, invariant and\imt^oodist(Rt{x),uj(x)) =0. PROOF
Relative compactness of j(x) implies that u(uo) is not empty.
Suppose zn E w(x) for n > 1. Pick tn > n such that d(Rtn(x),zn) < 1/n. Relative compactness of 7(0;) implies that a subsequence {Rtn. {x)} converges to some z E UJ(X). Clearly, zni —> z. This proves compactness of u)(x). Suppose there exist disjoint open sets A, B in M such that both u(x) fl A and LJ(X) fl B are nonempty and that LJ(X) C A U B. Then, for each n > 1 we
CHAPTER
278
6. SEMILINEAR
PARABOLIC
EQUATIONS
can find t2n-i > n and t2n > *2n-i such that Rt2n-i(x) £ ^ a n d Rt2n(x) e ^ Continuity of R.(x) implies that {Rt(x)}t2n-ii converges to some y. Now, this y should belong to both the closed set (AU B)c and UJ{X) - which is not possible. So, such sets A, B cannot exist and therefore u{x) is connected. Pick yo € w(x) and a sequence {tn}n>i such that tn —> oo and Rtn{x) -> yo as n —>• oo. Let Ao = N. For m > 1 let A m be an infinite subset of Am-\ such that {-Rt„-m(^)}neAm converges to some ym E w ( 4 Let no = 1 and for m > 1 pick n m G A m so that n m > n m _ i . Note that 72j G -Am for z > Tn and hence y m = lim Rtn.-m(x) l
i—too
for
all ra > 0.
Continuity of Rt+m^ Rt+k &nd the semigroup property imply Rt+m{ym) = lim J R t+m ( J R
l
i—too
l
whenever t > —ra and t > —k. This enables us to define u G C(M, u>(x)) by tz(£) = Rtjtm{ym) for £ > —m, ra > 0. This w has the required properties. If Rt(x) would not converge to u>(x), then there would exist c > 0 such that for each n there exists tn > n for which dist(Rtn(x),u(x)) > c. This is not possible because relative compactness of ^(x) would then imply that {Rtn{x)} would have a subsequence converging to some y G u(x). Therefore, Rt(x) converges to w(x) as £-» oo. □ We say that V G C(M, R) is a Liapunov function for Rt, t > 0, if (i) the mapping t —> V(Rt(x)) (ii) inf t > 0 V(Rt{x))
is nonincreasing on [0, oo) for every x G M
> - o o for each x G M
(iii) x G M is such that ^ ( ^ 0 * 0 ) = V(x) for t > 0 then x is an equilibrium point. The conditions ii) and iii) are sometimes omitted in the literature. However, one can sometimes find additional conditions attached. Theorem 6.8.3 If Rt, t>0, has a Liapunov function, then every nonempty iv-limit set consists of equilibrium points only. PROOF Let u(x) be nonempty and let V be the Liapunov function. Note that there exists c G R such that V(y) = c for all y G w{x). Pick yo G UJ(X) and a sequence {£n}n>i such that tn ->• oo and Rtn{x) ->> yo as n -» oo. Since i2t + t n (x) = Rt(Rtn(x)) ->- #t(j/ 0 ), we see that i^(y 0 ) € w(s) and hence V(i^(yo)) = c for all £ > 0. Thus, yo is an equilibrium point. □
6.9. EXAMPLE:
THE CHAFEE-INFANTE
PROBLEM
279
Rt E C(M, M), t > 0, is said to be a gradient system on M if it has a Liapunov function and if the orbit of every x E M is relatively compact. Various evolution equations generate gradient systems. Webb [32] has shown that a damped nonlinear wave equation is one such equation. A detailed analysis of this and various other gradient systems has been done by Hale in [10]. Here we shall now give an example of a semilinear parabolic equation that generates a gradient system.
6.9
Example: T h e Chafee-Infante Problem
We shall study the following PDE ut(x, t) = uxx(x, t) + g(u(x, t)) u{0,t)=u(£,t)=0 u(x,0) — uo(x)
for
0 < x < £, 0 < t < r,
for
0<*
for
0 < x < £,
while assuming that UQ E WQ(0,£) is real valued, g E C 1 (E, R) and that there exist a £ [0, TT2£-2) and b € R such that g(s)ds
G(x) = 2
x e R.
(6.53)
Jo As in Section 6.7, we are going to set the PDE as a semilinear parabolic equation in X = L 2 (0,^), with the linear operator A given by Av = -v"
vetD(A)
for
=
W2(0,£)nW^(0,£),
a e (1/4,1/2) and U = D(A Q ) = Xa. Theorem 6.4.10 implies existence of the solution u £ S(T) on some maximal interval [0, r ) . The solution is unique and real valued (see Theorems 6.4.6, 6.4.7, 6.7.1 and Corollary 6.7.3). For real valued v € WQ(0,£), define W{v) = /
{(v'(x))2 - G(v(x))) dx.
(6.54)
Jo L e m m a 6.9.1 W(u{-)) E C([0,r),R) and jW(u(t)) PROOF
= -2\\u'(t)\\2
forte
(0,r).
Choose 9 E (0,r) and t i , t 2 E [0,0].
Let h(t) = (Au(t),u(t))
= ^{ux{x,t))2dx.
Since A is self-adjoint we have
h(t2) - h(h) = (u(t2) - u(ii), Atx(*2)) + (Au(ti)Mh)
~ ti(*i)).
Hence the differentiability of u and the continuity of Au (Theorem 6.4.3) imply that ti = 2(u',Au) on (0,r). u E C([0,T),W£(0,£)) implies h E C([0,r),R).
CHAPTER 6. SEMILINEAR
280
Define k(t) = f£ G(u(x,t))dx /
/ 2(u(x,t2)
Jo Jo
PARABOLIC
EQUATIONS
and note that k(t2) - Hh) equals
-u(x,ti))g(u(x,ti)
+ 0{u(x,t2) - u(x,ti)))d0
dx.
Hence k{t2) - k(tx) = 2{u{t2) - ti(ti),p(ti(ti)) + etc). To estimate the etc term define c = maxo
an
d
no
^e that
\etc\ < \u(x,t2) — u(x, ti)| max \g'(s)\/2. \s\
Thus, k' = 2{u',g{u)) on (0,r) and k G C ( [ 0 , T ) , M ) . T h e o r e m 6.9.2 r = oo, sup||u(*)||i/ 2 < oo and inf W(u(t)) > -U. t>o *>° PROOF
(6.53) and Example 2.8.8 imply + a / u(x,t)2dx Jo
/ G(u(x,t))dx<M Jo
<M + a£27r~2 Jo
ux{x,t)2dx.
Hence (1 - a£2n~2) I ux(x, t)2dx
Jo
for
* G [0, r ) .
Since Lemma 6.9.1 implies that W(u(t)) < W(UQ), we have that ||w(t)||i/2 is bounded and hence g(u(t)) is bounded for t G [0, r ) . Theorem 6.4.10 implies that r = oo. □ Thus, if we define Rt(u0) = u(£), then Rt G C(W£{0,t),W£(0,£)), t > 0, is a dynamical system on W Q ( 0 , ^ ) . Moreover, W is a Liapunov function for Rt. Lemma 6.9.3 The orbit {u(t)}t>o is a relatively compact subset of
WQ(0,£).
PROOF Choose p G (1/2,1). Boundedness of g(u) (Theorem 6.9.2), Theo rem 6.1.15 and (6.15) imply that there exists c < oo such that \\Afiu{t)\\ < ct-Pe~rt + c [ (t- syPe-'^ds Jo
for
*> 0
where r G (0,7r 2 r 2 ) (recall that a (A) C [7r 2 r 2 ,oo)). Thus, {||^u(*)ll}t>i is bounded. Compactness of A - 1 (Example 2.8.3) implies that {Al^2~^A^u(t)}t>i is relatively compact in X (Exercise 5). Since the continuity implies that {A1/2w(^)}o
6.9. EXAMPLE:
THE CHAFEE-INFANTE
PROBLEM
281
Therefore, Rt, t > 0 is a gradient system on WQ(0,£) 6.8.3 imply
and Theorems 6.8.2 and
Theorem 6.9.4 U)(UQ), the u-limit set for UQ, is a nonempty, compact, connected subset of WQ (0, £). Moreover, in
A™
/ JM*)- ,y II 1/2 = 0
and if v £ U(UQ), then v G ?)(A) and Av — g(v).
Thus, to study the limiting behavior of u we need to study solutions of the steady state equation v"(x) + g{v(x)) = 0 for
x G [0, i],
v(0) = v{i) = 0.
(6.55)
Observe that if v satisfies (6.55) and if v'(xo) = 0 at some xo G (0,^), then v is symmetric around xo, i.e. if w(x) = v(2xo — x) then w = v near xo by the uniqueness of the initial value problem at XQ. Suppose now that the solution of (6.55) has a maximum M > 0. Choose the smallest XQ G (0, £) such that v(xo) = M. By continuity there should exist x\ G (0,#o) such that v > 0 on [xi,xo] and, by symmetry, v' > 0 on [xi,rco). Let X2 be the infimum of such x\. Note that v(x2) = 0 and v' > 0 on (#2,0:0). Multiplying (6.55) by v' and integrating gives {v')2 + G{v) = G{M).
(6.56)
Hence v'(x) = y/G(M) - G(v(x)) > 0 and rv(x)
L 0
ds
y/G{M) - G(s)
= X — X2
for
X G (X2,
XQ).
By symmetry, this determines v on [#2,2#o - # 2 ] C [0,£\. Note that ^'(0:2) > 0. When v'(x2) = 0 the symmetry implies that v is periodic with period 2(xo — ^2). Hence I = 2n(xo-x2) for some n G N. If x 2 > 0 and v ' ^ ) > 0, then G(0) < G(M) by (6.56) and v has a negative arch which can be analyzed similarly. The point 2xo — X2 can also be analyzed similarly. Let us gather some of these observations in the following: Theorem 6.9.5 Suppose that v is a solution of (6.55) and M = max v(x),
m = min v(x).
0<x
0<x
IfM>0, lfm<0, Moreover,
ds < 00. yJG(M)-G(s) ds then G{s) < G(m) for s G (ra,0) and £-(m) = / —j==^=== < 00. y/G(ri)~G(s) Jm
then G(s) < G(M) for s G (0,M) andi+(M)
= f /o Jo
282
CHAPTER
(1) IfM>Oandm
6. SEMILINEAR
= Q, then £ = 2n£+(M)
PARABOLIC
for some n G N .
(2) If M = 0 and m<0,
then £ = 2n£-(m)
(3) IfM>Oandm<0,
then G(M) = G(m) > G(0) and
£ = 2n+£+(M)
-f 2n-£-(m)
EQUATIONS
for some n G N.
for some n± G N with \n+ — n _ | < 1.
EXAMPLE 6.9.6 Let us take g(x) = Asinx where A > 0. Theorem 6.9.5 implies that any solution v of (6.55) satisfies — ir < v(x) < TT for x 6 [0,^]. Note that rA £+(A)
= U-A)
d<*
= /
for
A G (0,TT).
Jo v 2A(cos s — cos A) A change of variable 1 — cos s = (1 — cos A) cos 2 <^ gives that £+(A) = ^ = r VA7o
2
d{f
for
Ae(0,7r).
2
^l-sin (A/2)cosV
Hence, ^ + increases form 7r/\/4A at A = 0 to +oo at A = TT. If v ^ 0 satisfies (6.55) and A = max x |w(a;)|, then Theorem 6.9.5 implies that £ = 2n£+(A) > mr/y/X
for some
n G N.
Let N > 0 be the largest integer such that N < £\f\j'K. If n G N and n < N, then one can find a unique A G (0,7r) such that £ = 2n£+(A) and one can construct, as outlined above, two corresponding solutions of (6.55): one with v'(0) > 0 and one with v'(0) < 0. Since 0 is also a solution, we see that (6.55) has exactly 2N + 1 solutions for this g. Since cj-limit sets have to be connected (Theorem 6.9.4), we see that each consists of a single point. One can make a conjecture, based on t h e above T h e o r e m 6.9.5 a n d E x a m p l e 6.9.6, t h a t CJ(UO) always consists of a single point, which would imply t h a t t h e u always tends t o a solution of t h e steady state equation (6.55). This is actually a p o p u l a r m e t h o d a m o n g engineers for solving (6.55). This conjecture was proven u n d e r very mild assumptions on g. For more details a n d generalizations t o higher dimensions see Henry [11] a n d Hale [10].
6.10
Exercises
1. Suppose t h a t T —A is a n m-accretive operator on a Hilbert space for some A > 0. Show that T 1 / 2 can b e defined via Definition 6.1.6. Show t h a t T 1 / 2 is equal t o S as given in T h e o r e m 2.3.5. 2. For A G G(X), A G R, a G [0,1) a n d x G T>(Aa) show t h a t lim t-a\\x-extQ(t)x\\
= 0.
6.10. EXERCISES
283
3. Show that for every A G ®(X), x G X and a > 0 we have lim \a(A + \)-°lx
= x.
A->+oo
4. Show that for A G ®(X), a , f t 7 G l , a < ^ < 7 there exists c < oo such that \\Apx\\ < c | | A 7 a ; | | ^ | | A Q x | | ^
for all
x G V{A^).
5. Show that if A G ®(X) and A~a is compact for some a > 0, then A~P is compact for all /3 > 0. 6. Suppose that A —8 is an m-accretive operator in a Hilbert space for some 5 > 0. Show that Aa — 5a is an m-accretive operator for all a G [0,1]. 7. Let A be a linear operator in L 2 (0,^), t G (0, oo), given by Au = -u"
ueiD{A)
for
W2(0,i)nW^(0J).
=
Show that (a) T>{A1/2) = W£(0,e) and that \\Al/2u\\ = \\u'\\ for u G ^ ( 0 , ^ ) (b) for a > 1/4 there exists c < oo such that H^Hoo < c||AQu|| for u G T>(Aa). 8. Let A. be a sectorial operator in a Banach space X, a G (0,1) and a < Re A for all A G cr(A). Show that for each UQ £ X there exists a unique u G C([0, oo), X) such that w(0) = UQ and u'(£) + Au(£) = {A- a)au(t)
for all
£ G (0, oo).
9. Verify the details in Example 6.4.1 and construct an actual example for each 0 1/a. (Hint: try using multiplication operators.) 10. Show that T2
oo
oo
2 +m 2 )*\ ec _2(n -2(n-+m-)t^
n=l m=l
1/4
/
' ^ s i n t e
_ fc2 ,
«=1
satisfies, for every c > 0 ut(x,t)
= w xx (rr,t) + ( /
|t/x(-5, t)|2c?s j w(x,t)
u(0,t)=u{7r,t)=0 lim
for
sup |u(a5,£)| = 0.
t-¥0+ o
for t>0
t > 0, 0 < x < n
CHAPTER 6. SEMILINEAR PARABOLIC
284
EQUATIONS
(Hint: evaluate the Fourier sine series of ir —X and see Exercise 16 in Chapter 4.) Show that the PDE can be set as a semilinear parabolic equation in L 2 (0,7r), with a = 1/2 (see also Exercise 7) and that the uniqueness fails in this case because /
( /
\ux(x,t)\2dx\
eft = oo.
11. Show that Galerkin approximations of a solution of the Navier-Stokes equations exist globally, for any real valued initial UQ G X, and that they satisfy (6.41) on [0, oo). Prove that a subsequence converges weakly in L 2 ((0, oo) x 0 ) 3 . 12. Find the spectrum of the operator AQ in L 2 (0, oo) given by (6.45). 13. Prove Corollaries 6.7.4 and 6.7.5. 14. Suppose that g, in Section 6.7, is such that for some a, b G R we have g(x) < ax + b for
x> 0
and g(0) > 0. Show that if UQ > 0, then the solution of (6.50) - (6.52) exists for all time (r = oo). 15. Let Rt € C(M,M),
t > 0, be a dynamical system on M. Show that uj(x) = nT>o J(RT(X))
for all
x G M.
16. Suppose that UQ E C B ( R ) . Show that if
v(x,t) = -±= y/kltt
r
e-^-h!oMr)drds
J-oo
and u = —2vx/v, then u solves Burger's equation: ut(x,t) +u(x,t)ux(x,t)
= uxx(x,t)
u(x,0) = uo(x) This is known as Hopf-Cole substitution.
for
for x G R.
i 6 I , t > 0,
Bibliography [1] R. A. ADAMS, Sobolev Spaces, Academic Press, 1975. [2] S. N. CHOW AND K. LU, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74(1988), 285-317. [3] S. N. CHOW, D. R. DUNNINGER AND M. MIKLAVCIC, Galerkin Approxima tions for Singular Linear Elliptic and Semilinear Parabolic Problems, Applicable Analysis, 40(1990), 41-52. [4] P . CONSTANTIN AND C. FoiAS, Navier-Stokes Equation, The University of Chicago Press, 1988. [5] L. C. EVANS, Partial Differential Equations, Berkely Mathematics Lecture Notes, v. 3ab, 1994. [6] C. FoiAS, G. R. SELL AND E. S. TITI, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynamics and Diff. Eq., 1(1989), 199-244. [7] G. B. FOLLAND, Fourier Analysis and Its Applications, Brooks/Cole, 1992.
Wadsworth &
[8] AVNER FRIEDMAN, Partial Differential Equations, Krieger, 1983. [9] J. A. GOLDSTEIN, Semigroups of Linear Operators and Applications, Oxford University Press, 1985. [10] J. HALE, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, No. 25, AMS, 1988. [11] D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, No. 840, Springer, 1981. [12] L. HORMANDER, The Analysis of Linear Partial Differential Operators I, Springer, 1983.
285
BIBLIOGRAPHY
286
[13] T. KATO, Perturbation Theory for Linear Operators, 2nd edition, Springer, 1980. [14] H. KOMATSU, Fractional powers of operators, Pacific J. Math. 19(1966), 285346. [15] O. A. LADYZHENSKAYA, The Boundary Physics, Springer, 1985.
Value Problems of
Mathematical
[16] M. MIKLAVCIC, Nonlinear stability of asymptotic suction, AMS Transactions, 281(1984), 215-231. [17] M. MlKLAVClC, Stability for semilinear parabolic equations with noninvertible linear operator, Pacific J. Math. 118(1985), 199-214. [18] M. MlKLAVClC, Approximations for Weakly Nonlinear Evolution Math. Computations, 53(1989), 471-484.
Equations,
[19] M. MIKLAVCIC, A Sharp Condition for Existence of an Inertial Manifold, J. Dynamics and Diff. Eq., 3(1991), 437-456. [20] A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. [21] M. H. P R O T T E R AND H. F. WEINBERGER, Maximum Principles in Differential Equations, Springer, 1984. [22] J. RAUCH, Partial Differential Equations, Springer, 1991. [23] M. REED AND B. SIMON, Methods of Modern Mathematical Physics, v. I-III, Academic Press, 1972. [24] M. RENARDY AND R. C. ROGERS, An Introduction to Partial Differential Equa tions, Springer, 1993. [25] W. RUDIN, Functional Analysis, McGraw Hill, 1973. [26] W. RUDIN, Real & Complex Analysis, 3rd edition, McGraw Hill, 1987. [27] M. SCHECHTER, Principles of Functional Analysis, Academic Press, 1971. [28] M. SCHECHTER, Modern Methods in Partial Differential Equations, McGrawHill, 1977. [29] R. E. SHOWALTER, Hilbert Space Methods for Partial Differential Pitman, London, 1977.
Equations,
[30] R. TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMSNSF Regional Conference Series in Applied Mathematics, SIAM, 1983.
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[31] R. TEMAM, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. [32] G. W E B B , A Bifurcation Problem for a Nonlinear Hyperbolic Partial Differential Equation, SIAM J. Math. Anal. 10(1979), 922-932. [33] J. WLOKA, Partial Differential Equations, Cambridge University Press, 1987. [34] K. YosiDA, Functional Analysis, 6th edition, Springer, 1980.
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List of Symbols R real numbers C complex numbers K scalars (R or C) Z = {...,-1,0,1,2,...} N = {1,2,...} 0 empty set A closure or conjugate dist span{-} B(x,r) open ball C(M,N),CB{M,N) C(M,N),CH(M,N)
C(M) C0(M), supp(-) C m (fl), C°°{Sl) C^Q), C0°°(fi)
C?(a,b) C%(Q), || • |U,oo
A, Ju Bessel function DCT Je mollifier T Fourier transform S Schwartz space V(Q), V(Q) 1 7 1 4 4 7 7 7 8 8 9
cB{Q) = c°B(n)
c™(n) = cm{n)
Cu(il) = <7(fi) = C°(fi) C(, Cm
V, %o« II • Up #\co ©(•), *(•), *(') 2S(X,F),<8(X) AC[a,6] X* = 55(X,K) T* adjoint of T T* Hilbert space adjoint p(T), *(T), (7P(T) A A
10
D°> Di = a
£
D weak derivative 1 " \m,p
w%?, w£> w™*. ('•> '}m» II " l|m,p
L(S,X) 2l(a,M,0,X) e~Az
S m (r) 8(r) <3(X) AQ
*", II ' Ha w(0,7(-)
10 11 11 12 13 15 16 21 56 23 92 185 289
wm
193 95 78 110 115 114 122 7 127 133 135 135 172 193 196 218 260 247 249 254 277
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Index Abel's integral equation, 45 accretive, 28, 58, 73, 185 addition, 6 adjoint, 21, 56, 63 adjoint problem, 40 adjoint space, 16 almost separably-valued, 171 analytic, 27, 117, 169, 193 arcwise-connected, 5 Arzela-Ascoli Theorem, 5, 17, 31, 34, 90, 94, 150
compact resolvent, 37, 74, 78, 82, 83, 87, 90, 91, 94, 154, 155, 157 complement, 2 complete, 2, 48, 50, 74, 75, 78, 91, 92, 104, 163 complex vector space, 6 cone property, 141, 155 conjugate gradient method, 105 conjugate space, 16 connected, 5, 130, 155 continuous, 4 continuous at the point, 4 contraction, 3 Contraction Mapping Theorem, 3, 25, 216, 259 contraction semigroup, 184 converge, 2 convex, 7 convolution, 109, 123 cycle, 206
Baire Theorem, 2 Banach space, 9 Banach-Steinhaus Theorem, 14 basis, 7 Bessel function, 95 biharmonic operator, 156 Bochner integrable, 172 Bochner integral, 172 boundary value problem, 39, 120 bounded, 12 bounded linear functionals, 16, 55 bounded set, 9 Burger's equation, 284
defined in, 12 defined on, 12 delay equation, 257 delta function, 123 dense, 2 densely defined, 12 derivative of the distribution, 122 derivative of the product, 8 weak, 131 dimension, 7 Dirichlet problem, 153 discretization, 224 dissipative, 28
Cauchy sequence, 2 Cebysev polynomials, 52 closable, 15 closed, 2, 15 Closed Graph Theorem, 16 closure, 2, 16 compact, 2, 31 compact imbedding, 82, 83, 152, 156 291
292
distance, 1 distribution, 122, 127 domain, 11 dual space, 16 dynamical systems, 277 eigenvalues, 23 eigenvectors, 23 elliptic equation, 70, 96, 102, 118, 153 energy method, 271 entire, 117 equilibrium point, 277 equivalent norms, 9 Euler method, 183, 224, 235 extension, 12, 13, 18, 61, 99, 104 finite difference, 129, 234 finite element, 96, 107, 236, 237, 239 fixed-point, 3 formal adjoint, 63 Fourier cosine series, 52, 53 Fourier series, 52, 75, 104 Fourier sine series, 52, 79 Fourier transform, 114, 115, 120, 121, 129, 137, 159 fractional powers of operators, 58, 249 Fredholm alternative, 38, 42, 57, 83, 156, 157 Friedrichs extension, 99, 243 fundamental solution, 123 Galerkin approximation, 98, 99, 103, 236, 239, 243 Garding's inequality, 103, 156, 157 generator, 168 gradient system, 279 Gram-Schmidt orthogonalization, 48 Gronwall Theorem, 216, 258 Hahn-Banach Theorem, 18 Hamiltonian, 88, 187 harmonic functions, 120 hat function, 98
INDEX
Hausdorff Maximality Theorem, 18 heat equation, 222, 260, 267 Heisenberg inequality, 117 Hermite functions, 91, 163 Hilbert space, 48 Hilbert space adjoint, 56 Hilbert-Schmidt Theorem, 74 Hille-Yosida Theorem, 180 Holder continuous, 4 Hopf-Cole substitution, 284 Hormander Theorem, 70 hyperbolic equation, 66, 187 imbedding compact, 82, 83, 152, 156 infinitesimal generator, 168 initial value problem, 120, 121 inner product, 47 inner product space, 47 instability, 201, 265 intermediate derivatives, 132 interpolation inequalities, 132, 135 invariant, 277 invariant subspace, 204 inverse Fourier transform, 115 Laplace operator, 70 Laplace transform, 116 Laplacian, 70, 92, 118, 153, 155, 163, 185, 193 Legendre polynomials, 48, 104 Liapunov function, 278, 280 linear operator from X to F , 11 linearly independent, 7 Lipschitz continuous, 4 m-accretive, 58, 61, 73, 87, 186, 283 Malgrange-Ehrenpreis Theorem, 125 maximal totally ordered, 18 maximum principle, 275 Maxwell's equations, 164 metric, 1 metric space, 1 mild solution, 210, 234, 242
INDEX
mild solutions, 218 mollifier, 110 Neumann problem, 155, 165 norm, 8 normalized, 47 normalized tangent functional, 20, 172 normed space, 8 null space, 12 numerical range, 28, 80, 99, 192 u;-limit set, 277 open, 2 open ball, 1 Open Mapping Theorem, 14 orbit, 277 origin, 6 orthogonal, 47 orthogonal projection, 55 orthonormal, 48 orthonormal basis, 48 parabolic equation, 66, 120, 169, 185, 192, 200, 222, 233, 236, 237, 243 Parseval equality, 50, 52 partially ordered, 18 Partition of Unity Theorem, 110 perturbation of a generator, 186 Pettis Theorem, 171 point spectrum, 23 Ponicare inequality, 138, 154, 165 positive definite, 67 positively invariant, 277 projection, 55, 205 projection associated with the spectral set, 206 range, 12 rapidly decreasing functions, 114 real subspace, 7, 174, 205, 220, 263 real vector space, 6 reflexive, 20
293
regularity of weak solutions, 64, 160 relatively compact, 2 resolvent, 23 resolvent identity, 23 resolvent set, 23 restriction, 12, 61 restriction of A to Y, 204 retarded functional differential equation, 257 retraction map, 217 Riesz Lemma, 55 scalar, 6 scalar field, 6 scalar multiplication, 6 scalar product, 47 Schrodinger equation, 187 Schwartz space, 114 second dual space, 20 sectorial operator, 193, 201, 253, 254, 258, 259, 283 sectorial sesquilinear forms, 80, 89, 93, 97, 99, 153, 155-157, 186, 187, 193, 236, 239, 253, 273 self-adjoint, 57, 71, 73-75, 78, 83, 84, 88, 91, 93, 94, 99, 105, 107, 154-156 semigroup, 167 analytic, 192 semilinear parabolic equations, 260 seminorm, 9 separable, 2, 5, 11, 17 simple functions, 173 singular differential equation, 103 smoothing operator, 170, 201 span, 7 spectral representation, 74 spectral set, 206, 213 spectrum, 23 square root, 58, 87 stability, 201, 264, 271 stability condition, 225
294
star-shaped, 138, 165 strong ellipticity condition, 156, 157, 200 strongly continuous semigroup, 167 strongly elliptic operator, 157 strongly measurable, 171 Sturm-Liouville problem, 76 subspace, 7 support, 7 symmetric, 57, 70 symmetric hyperbolic system, 66 Taylor formula, 8 totally ordered, 18 Trotter Theorem, 226 uncertainty principle, 118 uniform boundedness principle, 14, 17 uniformly continuous, 4 uniqueness, 260, 283 variational formulation, 85 vector space over , 6 Volterra integral equation, 26 wave equation, 68, 121, 189, 239, 243, 246, 279 generalized, 170, 187, 190 weak derivative, 127 weak limit, 18 weak solution, 63 weakly bounded, 44 weakly convergent, 18 weakly measurable, 171 weakly nonlinear evolution equations, 215 Young Lemma, 112 zero vector, 6
INDEX
