Global existence for the heat equation with nonlinear dynamical boundary conditions

GLOBAL EXISTENCE FOR THE HEAT EQUATION WITH NONLINEAR DYNAMICAL BOUNDARY CONDITIONS ENZO VITILLARO Abstract. The paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is 8 > ut − ∆u = 0 in (0, ∞) × Ω, > > > > = −|ut | ut + |u| u on [0, ∞) × Γ1 , > > > ∂ν > :u(0, x) = u (x) on Ω, 0

R

where Ω ⊂ n (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪Γ1 , m > 1, 2 ≤ p < r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n = 1, 2, and u0 ∈ H 1 (Ω), u0 = 0 on Γ0 . We prove local existence of the solutions in H 1 (Ω) when m > r/(r + 1 − p) or n = 1, 2, and global existence when p ≤ m or the initial data are inside the potential well associated to the stationary problem.

1. Introduction and main results We consider the problem  ut − ∆u = 0     u = 0 (1) ∂u  = −Q(t, x, ut ) + f (x, u)     ∂ν u(0, x) = u0 (x)

in (0, ∞) × Ω, on [0, ∞) × Γ0 , on [0, ∞) × Γ1 , on Ω,

where u = u(t, x), t ≥ 0, x ∈ Ω, ∆ denotes the Laplacian operator, with respect to the x variable, Ω is a bounded open subset of Rn (n ≥ 1) of class C 1 (see [4]), ∂Ω = Γ0 ∪ Γ1 , Γ0 ∩ Γ1 = ∅, Γ0 and Γ1 are measurable over ∂Ω, endowed with the (n − 1) – dimensional Lebesgue measure λn−1 . These properties of Ω, Γ0 and Γ1 will be assumed, without further comments, throughout the paper. The initial datum u0 belongs to H 1 (Ω) and u0 = 0 on Γ0 . Moreover Q represents a nonlinear boundary damping term, i.e. Q(t, x, v)v ≥ 0, and f represents a nonlinear source term, i.e. f (x, u)u ≥ 0. Local and global existence for the solutions of problems like (1) has been widely studied when Q ≡ 0 (parabolic problems with nonlinear boundary conditions) or Q ≡ ut (parabolic problems with dynamical boundary conditions). We respectively refer to [2], [8], [16], [17], [18] and to [6], [9], [10], [11], [14], [15]. The quoted papers contain, roughly, three different kinds of results: local existence (with various regularity assumptions on u0 and f ), global existence when f Work done in the framework of the M.U.R.S.T. project ”Metodi variazionali ed equazioni differenziali nonlineari”. 1

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ENZO VITILLARO

is sublinear in u for |u| large or when λn−1 (Γ0 ) > 0 and u0 is small in a suitable sense, via the so-called potential well theory, and finally, blow-up when u0 is large enough and f is superlinear in u for |u| large. In this paper we study the case Q(t, x, ut ) ' |ut |m−2 ut , m > 1 and f (x, u) ' p−2 |u| u, p > 2. For the sake of simplicity we shall consider in the sequel the model problem  ut − ∆u = 0 in (0, ∞) × Ω,     u = 0 on [0, ∞) × Γ0 , (2) ∂u  = −|ut |m−2 ut + |u|p−2 u on [0, ∞) × Γ1 ,     ∂ν u(0, x) = u0 (x) on Ω. When the term |ut |m−2 ut is not present in (2) and λn−1 (Γ0 ) > 0, this type of problem was considered in [18]. In particular global existence was proven when the datum u0 belongs to a suitable stable set W (see (13)) using potential well arguments. When m = 2 problem (2) can be considered as a particular case in the general theory developed in [10] and [11], where semigroup theory arguments were applied. In particular local existence and uniqueness of a maximal solution u ∈ C 0,1 ([0, Tmax ); H 1 (Ω)) ∩ C 0,δ ([0, Tmax ); H 1,s (Ω)) (where δ = (1 − s)/2, s ∈ [0, 1)) were proven when u0 ∈ H 1 (Ω), Γ0 = ∅ and 2 ≤ p < 2(n − 1)/(n − 2) or n = 1, 2. Moreover a blow–up result was given when 2 < p < 2(n − 1)/(n − 2) or n = 1, 2 and u0 is large enough. Some related problems concerning wave equations with nonlinear damping and source terms have been considered in [13], [24], [27]. In particular [13] deals with the Cauchy–Dirichlet problem  m−2  ut = |u|p−2 u in (0, ∞) × Ω, utt − ∆u + |ut | (3) u=0 on [0, ∞) × ∂Ω,   u(0, x) = u0 (x), ut (0, x) = u1 (x) on Ω, when m ≥ 2 and 2 < p ≤ 1 + 2∗ /2 (2∗ denotes the critical exponent of the embedding H 1 (Ω) ,→ Lp (Ω)), while [24] deals with the Cauchy problem associated to (3) when Ω is replaced by the entire space Rn . In [24] the more general case m > 1, 2 ≤ p < 2∗ , m > 2∗ /(2∗ + 1 − p) was considered for the first time. In ∂ [27] the author studied a modified version of (1) where the heat operator ∂t − ∆ is 2 ∂ replaced by the d’Alambertian operator ¤ = ∂t2 − ∆ and 2 ≤ p < 2(n − 1)/(n − 2) or n = 1, 2. The first aim of the present paper is to show that, when 2 ≤ p ≤ m and p < 2(n − 1)/(n − 2) or n = 1, 2, the solutions of (2) are global in time, for arbitrary (even large) initial datum, so that the blow–up phenomenon shown when Q ≡ 0 (see [8] and [16]), and when Q ≡ ut (see [11] and [15], dealing with a small perturbation of (2)) cannot occurs. This type of results was proved by the author in [27] in connection with wave type equations. In general methods employed to study hyperbolic problems cannot be employed to study parabolic problems, and conversely. Nevertheless, the arguments of [27] can be conveniently adapted to problem (2). However there are several important differences in the proofs, which make the adaptation non–trivial and

3

which will be outlined at the end of this section. Moreover, the extension of these results to (2) enlightens the fact that the superlinear boundary damping term is the main factor who determines the life–span of the solutions, regardless to the parabolic or hyperbolic structure of the differential operator acting on Ω. A natural conjecture arising from this result is that the wave operator can be replaced by the Laplacian operator too. To state our results we set   2(n − 1) if n ≥ 3, (4) r= n−2 ∞ if n = 1, 2, to be the critical value of the trace – Sobolev embedding H 1 (Ω) ,→ Lp (∂Ω). Moreover we introduce the notations k · kq = k · kLq (Ω) ,

k · kq,Γ1 = k · kLq (Γ1 ) ,

1≤q≤∞

and HΓ10 (Ω) = {u ∈ H 1 (Ω) : u|Γ0 = 0},

kuk2H 1

Γ0 (Ω)

= kuk22 + k∇uk22 .

For the sake of simplicity we state our main results only in the case n ≥ 3. The first step in our study is the following local existence result. Theorem 1. Let m > 1, (5)

2≤p
Then, given u0 ∈ (0, T ) × Ω such that

and

m > r/(r + 1 − p).

there is T > 0 and a weak solution u

(6)

u ∈ C([0, T ]; HΓ10 (Ω)),

(7)

ut ∈ L2 ((0, T ) × Ω) ∩ Lm ((0, T ) × Γ1 ),

1

of (2) on

and the energy identity Z tZ ¯t Z t ¡ ¢ 1 m 2¯ 2 (8) |u|p−2 uut k∇uk2 ¯ + kut k2 + kut km,Γ1 = 2 s s Γ1 s holds for 0 ≤ s ≤ t ≤ T . Moreover T = T (ku0 k2H 1 the first variable.

Γ0 (Ω)

, m, p, Ω, Γ1 ) is decreasing in

The assumption m > r/(r +1−p) in Theorem 1 cuts a part of the full subcritical range p < r and m > 1 (see Figure 1). This type of condition appears in [13], [24] and [27] too, as recalled above, and depends on the presence of the nonlinear damping term |ut |m−2 ut together with the low regularity required for u0 . When m = 2 and Γ0 = ∅ this restriction is not necessary, as shown in [11]. On the other hand problem (2) posses in this case a very particular structure which allows to apply semigroup arguments, since it can be written as an abstract Cauchy problem of the form z 0 (t) + A(z(t))z(t) = F (z(t)), z(0) = z0 , by setting z = (u, u|Γ1 ). An approach of this type seems very problematic when m 6= 2. Let us explain briefly were the assumption m > r/(r + 1 − p) comes from. To handle with |ut |m−2 ut we use the classical monotonicity method (see [20] or 1The precise definition of weak solution will be given in Section 3.

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[21]), in which the energy identity (8) plays an essential role. Since u0 ∈ H 1 (Ω) the best regularity of the solution u that one can expect (at lest when m 6= 2) is that given by (6). Furthermore, a – priori boundary estimates on ut only give ut ∈ Lm ((0, T ) × Γ1 ). Then, in view of the trace – Sobolev embedding H 1 (Ω) ,→ Lp (∂Ω) and of H˝older inequality, the integral in the right hand side of (8) makes sense only if m ≥ r/(r + 1 − p). To explain why the inequality m ≥ r/(r + 1 − p) is assumed in its stronger form we have to give some detail from the proof. A standard contraction argument is not available and a compactness argument (Schauder fixed point theorem) is instead used. This immediately explains why the case p = r cannot be considered, since in this case the embedding H 1 (Ω) ,→ Lp (∂Ω) is not compact. Moreover, since the estimate of the nonlinear damping term plays an essential role, we cannot apply a compactness arguments when m = r/(r + 1 − p). A further consequence of the method applied is that we cannot assert uniqueness of the solutions of (2), as in [24] and [27]. We cannot use a contraction argument even with some further (non– trivial) restrictions on the parameters p and m, as in[13], since here we cannot estimate the boundary damping term using the L2 norm. Our global existence result is the following Theorem 2. Let (9)

2 ≤ p ≤ m,

and

p < r.

Then any weak solution given by Theorem 1 can be extended to the whole of (0, ∞)× Ω. Theorem 2 is the first result, in author’s knowledge, which shows that a superlinear boundary damping term forces global existence for the solutions of the heat equations with source terms, with arbitrary initial data. Since in [13] the condition p ≤ m was proved to be optimal to ensure the global existence of the solutions of (3), we conjecture here that the same phenomenon happens for (2), i.e. that when p > m solutions blow–up in finite time. The arguments of [11] show that this conjecture is correct when m = 2. 2 The second aim of the paper is to extend the potential well theory result of [18] to (2). The stable set we consider is the same introduced in [18]–[19]. To recall it we set 1 1 (10) J(u) = k∇uk22 − kukpp,Γ1 , 2 p and (11)

K(u) = k∇uk22 − kukpp,Γ1

defined on HΓ10 (Ω). When λn−1 (Γ0 ) > 0 Poincar`e inequality holds (see [28]), so that k∇uk2 is an equivalent norm on HΓ10 (Ω), and the number (12)

d=

inf

sup J(λu)

1 (Ω),u| u∈HΓ Γ1 6≡0 λ>0 0

2This blow–up result is stated in [11] for a vectorial version of (2) when Γ = ∅, but the 0 arguments there can be applied without essential changes to the scalar case and when Γ0 6= ∅

5

Figure 1. The picture describes the three regions in the (p, m) plane related to the result we obtained for problem (2). In region I global existence for arbitrary initial data is proved (Theorem 2). In region II local existence for arbitrary initial data (Theorem 1) and global existence for initial data in the potential well (Theorem 3) when λn−1 (Γ0 ) > 0 is obtained. On the segment III blow–up for a large datum was proven in [11]. is positive (see [19] or Lemma 3 below, where a different characterization of d is given). The stable set W of the initial data, for which global existence is proven in [18], is defined by (13)

W = {u0 ∈ HΓ10 (Ω) : K(u0 ) ≥ 0

and

J(u0 ) < d}.

We can now state Theorem 3. Suppose that m > 1, 2≤p
m > r/(r + 1 − p),

and λn−1 (Γ0 ) > 0. If u0 ∈ W , then there is a global solution u of (2) on (0, ∞) × Ω such that u(t) ∈ W for all t ≥ 0, (14)

u ∈ C([0, ∞); HΓ10 (Ω)),

(15)

ut ∈ Lm ((0, ∞) × Γ1 ) ∩ L2 ((0, ∞) × Ω).

and the energy identity (16)

¯t 1 ¯ k∇uk22 ¯ + 2 s

Z s

t

2 kut km m,Γ1 + kut k2 =

Z tZ s

|u|p−2 uut Γ1

holds for 0 ≤ s ≤ t < ∞. Remark 1. It is easy to see that the set W is bounded in HΓ10 (Ω), so that the solution u also belongs to L∞ (0, ∞; HΓ10 (Ω)).

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ENZO VITILLARO

As a byproduct of the proof of Theorem 1 we also obtain the following global existence–uniqueness result for the problem  ut − ∆u = 0 in (0, T ) × Ω,     u = 0 on [0, T ) × Γ0 , (17) ∂u  = −|ut |m−2 ut + g(t, x) on [0, T ) × Γ1 ,     ∂ν u(0, x) = u0 (x) on Ω, where m > 1, T > 0 is arbitrary and g is a given forcing term acting on Γ1 . 0

Theorem 4. Suppose that g ∈ Lm ((0, T ) × Γ1 ), where 1/m + 1/m0 = 1. If u0 ∈ HΓ10 (Ω) then there is a unique weak solution 3 u of (17) on (0, T ) × Ω. Moreover (18)

u ∈ C([0, T ]; HΓ10 (Ω)),

(19)

ut ∈ Lm ((0, T ) × Γ1 ) ∩ L2 ((0, T ) × Ω)

and the energy identity (20)

Z tZ ¯t Z t ¡ ¢ 1 2 m 2¯ kut k2 + kut km,Γ1 = gut k∇uk2 ¯ + 2 s s s Γ1

holds for 0 ≤ s ≤ t ≤ T . Theorems 1 – 3 are extended in Section 4 to problem (1), under suitable assumptions on the nonlinearities f and Q. For the sake of clearness we first present the proofs for the model problem (2) and then we give in (4) the necessary generalizations needed to handle with (1). This section is naturally addressed to a more specialized audience and then an higher lever of mathematical expertise of the reader is supposed. In particular the proofs are only sketched. In order to explain the main difficulties which arise in the proofs of Theorems 1– 4 we now make some comparison with the arguments used in [27]. Theorem 4, which is the parabolic version of [27, Theorem 4], is proved via Faedo–Galerkin procedure, so that finite dimensional approximations of (17) are considered. The first essential difference, with respect to [27], comes out here, since the boundary damping appearing in (17) is now a time derivative of the same (highest) order of ut acting on Ω. Moreover, different a–priori estimates are used in the proof of Theorem 4. Theorem 1 is proven by using Theorem 4 together with a fixed point argument. The main estimates of [27] are here conveniently modified to handle with (2). It is worth to mention that the energy identity (20) used in the proof of Theorem 1 differs from the usual energy identity used for heat equation (see for example [12]) since it is obtained by formally multiplying the equation by ut as for wave type equations instead that by u. This gives a more precise explanation of why the transposition of techniques motivated by hyperbolic problems gives an original outcomes when it can be applied to parabolic problems. The proofs of Theorems 2 and 3 do not essentially differ from the analogous ones given in [27], but they are explicitly given because of their simplicity and for the sake of clearness. Finally we wold like to mention that the techniques used in the paper also allow to consider modified versions of problem (1) where the Laplacian operator ∆ is replaced by more general second order uniformly elliptic operators in divergence form, such as 3The precise definition of weak solution will be given in Section 2.

7

Au = div(A0 (x)∇u), where A0 (x) = (aij (x))i,j=1,··· ,n , aij ∈ L∞ (Ω), i, j = 1, · · · , n where v t A(x)v ≥ α0 |v|2 for all v ∈ Rn and some positive constant α0 . Theorems 1 – 3 could also be extended to the vectorial case of systems of the form (1), using the same arguments. These generalizations are left to the interested reader. Notations and preliminaries Let us collect some notation and preliminaries that we shall use during the paper: q0

H˝older conjugate of q > 1, i.e. 1/q + 1/q 0 = 1,

(·, ·)

scalar product in L2 (Ω),

Cc∞ (O)

space of compactly supported C ∞ function on O,

C([a, b]; X)

space of strongly continuous functions from [a, b] to X,

Cc ((a, b); X)

space of compactly supported strongly continuous functions from (a, b) to X,

Cw ([a, b]; X) X

0

space of weakly continuous functions from [a, b] to X, the dual space of X,

h·, ·iX

the duality product between X and X 0 ,

H −1 (Ω)

the dual space of H01 (Ω),

L(X, X 0 )

the space of linear bounded operators from X to X 0 ,

where X is a Banach space and O any open subset of Rk . Moreover we shall call the trace theorem the continuous embedding HΓ10 (Ω) ,→ L2 (Γ1 ). We shall also refer as the trace–Sobolev embedding to the continuous embedding (see [1]) HΓ10 (Ω) ,→ Lp (Γ1 )

for 2 ≤ p < r.

2. Global existence and uniqueness for a boundary forced heat equation This section is devoted to the proof of Theorem 4. We first give the following Definition 1. A weak solution of (17) is a function u such that: a) u ∈ L∞ (0, T ; HΓ10 (Ω)), ut ∈ L2 ((0, T ) × Ω), b) the trace of u on (0, T ) × ∂Ω (which exits by the trace theorem) has a distributional time derivative on (0, T ) × ∂Ω, belonging to to Lm ((0, T ) × ∂Ω), c) for all (21)

φ ∈ X := {u ∈ HΓ10 (Ω) : u|Γ1 ∈ Lm (Γ1 )},

(22)

and for almost all t ∈ [0, T ], we have Z Z Z ut (t)φ + ∇u(t)∇φ + |ut (t)|m−2 ut (t)φ = Ω

d) u(0) = u0 .

Γ1

Γ1

g(t)φ,

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ENZO VITILLARO

Note that, in d), u(0) makes sense since by a) u ∈ W 1,2 (0, T ; L2 (Ω)) ,→ C([0, T ]; L2 (Ω)). In the proof of Theorem 4 we shall use the following Lemma 1. Let 0

ζ ∈ Lm ((0, T ) × Γ1 ),

(23)

and suppose that u is a weak solution of   u − ∆u = 0   t u=0 (24)    ∂u = ζ ∂ν i.e. a function

in (0, T ) × Ω, on [0, T ) × Γ0 , on [0, T ) × Γ1 ,

u ∈ L∞ (0, T ; HΓ10 (Ω)),

(25) such that (26) and

ut ∈ L2 (0, T ; L2 (Ω)) ∩ Lm ((0, T ) × Γ1 ), Z

Z

(27)

ut (τ )φ + ∇u(τ )∇φ = Ω

ζ(τ )φ = 0 Γ1

for all φ ∈ X and almost all τ ∈ [0, T ]. Then u ∈ C([0, T ]; HΓ10 (Ω))

(28) and the energy identity (29)

¯t 1 ¯ k∇uk22 ¯ + 2 s

Z

t s

kut k22 =

Z tZ ζut s

Γ1

holds for 0 ≤ s ≤ t ≤ T . Lemma 1 is an extension of [25, Theorems 3.1 and 3.2], which cannot be directly applied to (24). Nevertheless the technique of [25] continues to work in the present situation. Since the proof of this result is rather technical, we give it in Appendix A for the reader convenience. We can now give the Proof of Theorem 4. We apply the Faedo–Galerkin procedure. Let (wk )k be a sequence of linearly independent vectors in X whose finite linear combinations are dense in X. By (eventually) using the Grahm–Schmidt orthogonalization process we can take (wk )k to be orthonormal in L2 (Ω). Since X is dense in HΓ10 (Ω) (see Pk j for example [27, Appendix A]) there are u0k = j=1 y0k wj , k ∈ N, for some real j numbers y0k , j = 1, . . . , k, k ∈ N, such that (30)

u0k → u0

in HΓ10 (Ω).

For any fixed k ∈ N we look for approximate solutions of (17), that is for solutions Pk uk (t) = j=1 ykj (t)wj of the finite dimensional problem ( R R (ukt , wj ) + (∇uk , ∇wj ) + Γ1 |ukt |m−2 ukt wj = Γ1 gwj , j = 1, . . . , k, (31) k u (0) = u0k .

9

In order to recognize that (31) has a local solution we set 1 k t y0k = (y0k , . . . , y0k ),

yk = (yk1 , . . . , ykk )t ,

Ak = ((∇wi , ∇wj ))i,j=1,...,k , Bk (x) = (w1 (x), . . . , wk (x))t , Z |Bk (x) · y|m−2 Bk (x) · yBk (x) dx, y ∈ Rk , Gk (y) = y + Γ1

and (32)

Z Hk (t) =

g(t, x)Bk (x) dx. Γ1

Problem (31) can be rewritten in the vectorial form ( Gk (yk0 (t)) + Ak yk (t) = Hk (t), (33) yk (0) = y0k . Since wj ∈ Lm (Γ1 ) for j = 1, . . . , k we have |Bk | ∈ Lm (Γ1 ) and Gk ∈ C(Rk ). Moreover Gk = ∇Gk , where Z 1 1 Gk (y) = |y|2 + |Bk (x) · y|m dx. 2 m Γ1 The first addendum in previous formula is strictly convex, while the second one is convex since Z ¯ ¯ 1 ¯Bk (x) · [αy1 + (1 − αy2 ]¯m dx m Γ1 Z ¯ ¯ 1 ¯αBk (x) · y1 + (1 − α)Bk (x) · y2 ]¯m dx = m Γ1 Z ¯ ¯m ¯ ¯m 1 ≤ α¯Bk (x) · y1 ¯ + (1 − α)¯Bk (x) · y2 ¯ dx, m Γ1 for any α ∈ (0, 1), where the convexity of the function |y|m was used. Then Gk is strictly convex in Rk . Moreover, for any y 6= 0, Gk (λy) = ∞. λ→+∞ λ lim

Then, using [22, Theorems 2.5 and 2.6], we recognize that Gk is an homeomorphism from Rk onto itself, with inverse G−1 k . Then (33) can be written in the form ( yk0 (t) = G−1 k (Hk (t) − Ak yk (t)), (34) yk (0) = y0k . Now we note that, since Gk (y)y ≥ |y|2 for all y ∈ Rk , by Schwartz inequality it k follows that |y| ≤ |Gk (y)|. Then |G−1 k (y)| ≤ |y| for all y ∈ R , so that (35)

|G−1 k (Hk (t) − Ak yk )| ≤ |Hk (t)| + kAk k|yk |. 0

Since |Bk | ∈ Lm (Γ1 ) and g ∈ Lm ((0, T ) × Γ1 ) it follows that Hk ∈ L1 (0, T ). We can then apply Carath´eodory’s theorem ([5, Theorem 1.1]) to conclude that (34), and then (33) and (31), have a local solution on (0, tk ) for some tk > 0.

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Multiplying (31) by (ykj )0 and summing for j = 1, . . . , k we obtain the energy identity (here and in the sequel explicit dependence on t will be omitted, when clear) µ ¶ Z d 1 k 2 k 2 k m (36) k∇u k2 + kut k2 + kut km,Γ1 = gukt . dt 2 Γ1 Integrating over (0, t), 0 < t < tk , and using Young’s inequality, we obtain Z t ¢ ¡ k 2 1 kut k2 + kukt km k∇uk k22 + m,Γ1 2 0 Z 0 1 1 1 t k m ≤ k∇u0k k22 + 0 kgkm kut km,Γ1 . + 0 Lm ((0,T )×Γ1 ) 2 m m 0 Then using (30) there is K1 = K1 (k∇u0 k2 , kgkLm0 ((0,T )×Γ1 ) , m) > 0 such that k∇uk kL∞ (0,tk ;L2 (Ω)) kukt kL2 ((0,tk )×Ω) kukt kLm ((0,tk )×Γ1 ) k m−2 k k|ut | ut kLm0 ((0,tk )×Γ1 )

(37)

≤ K1 , ≤ K1 , ≤ K1 , ≤ K1

for k ∈ N. By (30), (37) and H˝older inequality in time it follows that µZ tk ¶1/2 Z t (38) kuk k22 ≤ ku0k k2 + kukt k2 ≤ ku0k k2 + T 1/2 kukt k22 ≤ K2 0

for some K2 = K1 (ku0 kHΓ1

0

0

(Ω) , kgkLm0 ((0,T )×Γ1 ) , T, m) > 0. Since (wk )k is orthonor1/2

mal in L2 (Ω), we have |yk (t)| = kuk (t)k2 , so (38) yields that |yk (t)| ≤ K2 . Then, by (35) 1/2 ∈ L1 (0, T ). |G−1 k (Hk (t) − Ak yk )| ≤ |Hk (t)| + kAk kK2 We can then apply [5, Theorem 1.3] to conclude that tk = T for k = 1, . . . , n. Next, by (37) and (38) it follows (see [20] and [21]) that, up to a subsequence, uk → u

weakly∗ in L∞ (0, T ; HΓ10 (Ω)),

ukt → ut

weakly in L2 ((0, T ) × Ω),

ukt → ψ

weakly in Lm ((0, T ) × Γ1 ),

|ukt |m−2 ukt → χ

weakly in Lm ((0, T ) × Γ1 ).

(39)

0

A consequence of the convergence (39) is that uk → u weakly in H 1 (0, T ; L2 (Ω)) and then, up to a subsequence, uk → u strongly in C([0, T ]; L2 (Ω)), so that u(0) = u0 . It follows in a standard way (see for example [27, p. 272], that ψ is the distribution time derivative of u on (0, T ) × ∂Ω, i.e. ψ = ut . Next, multiplying (31) by φ ∈ Cc∞ (0, T ), integrating on (0, T ), passing to the limit as k → ∞ (using (39)), and finally using the density of the finite linear combinations of (wk )k in X we obtain ¸ Z T· Z Z (40) (ut , w)φ + (∇u, ∇w)φ + χwφ − gwφ = 0 0

for all w ∈ X, φ ∈

Γ1

Cc∞ (0, T ),

and then

Γ1

Z

Z

(ut , w) + (∇u, ∇w) +

χw = Γ1

gw Γ1

11

almost everywhere in (0, T ). Then, to prove that u is a weak solution of (17) we have only to show that χ = |ut |m−2 ut

(41)

a.e. on (0, T ) × Γ1 .

By Lemma 1 we obtain (18) and the energy identity (29), with ζ = g − χ, which for s = 0 and t = T reads as Z TZ Z TZ ¯T Z T 1 2¯ 2 kut k2 + χut = gut . (42) k∇uk2 ¯ + 2 0 0 0 Γ1 0 Γ1 The classical monotonicity method then (see [20] and [21]) then allows to prove (41) together with the uniqueness of the solution. We briefly outline here the proof of these facts, for the reader convenience. Integrating (36) over (0, T ), extracting a subsequence such that ∇uk (T ) → ∇u(T ) weakly in L2 (Ω), passing to the limit as k → ∞, using (39) and the weakly lower semi–continuity of norms in Hilbert spaces we get Z T Z T Z TZ 1 1 2 2 k m 2 (43) k∇u(T )k2 + kut k2 + lim kut km,Γ1 ≤ k∇u0 )k2 + gut . 2 2 k 0 0 0 Γ1 By combining (42) with (43) we have Z T Z lim kukt km ≤ m,Γ1 k

0

0

T

Z χut . Γ1

The last formula, together with (39) and the monotonicity of the map y → |y|m−2 y gives Z TZ Z TZ ¡ k m−2 k ¢ k m−2 0 ≤ lim |ut | ut − |φ| φ (ut − φ) ≤ (χ − |φ|m−2 φ)(ut − φ) k

0

Γ1

0

Γ1

m

for all φ ∈ L ((0, T )×Γ1 ). Setting φ = ut −λφ1 , λ > 0, letting λ → 0+ and dividing RT R by λ we get 0 Γ1 (χ − |ut |m−2 ut )φ1 ≥ 0 for all φ1 ∈ Lm ((0, T ) × Γ1 ), and then (41) follows. The uniqueness of the solutions follows applying (29) to the difference w=u ˜−u ˆ of any two solutions (which solves (24) with ζ = −|˜ ut |m−2 u ˜t + |ˆ ut |m−2 u ˆt and w(0) = 0) and using the monotonicity of |y|m−2 y. ¤ 3. Proof of main results In this section we prove Theorems 1–3. We first give the precise definition of weak solution of (2). Definition 2. A weak solution of (2) is a function u such that a) –d) of Definition 1 hold, with (22) replaced by Z Z Z m−2 (44) ut (t)φ + ∇u(t)∇φ + |ut (t)| ut (t)φ = |u(t)|p−2 u(t)φ. Ω

Γ1

Γ1

To prove Theorem 1 we use the following compactness result. The proof can be found in [27]. Lemma 2. Let m > 1 and 1 < p0 < r. Set ZT := {u ∈ L∞ (0, T ; HΓ10 (Ω)) : ut ∈ Lm ((0, T ) × Γ1 )}

12

ENZO VITILLARO

endowed with the natural norm kuk2ZT = kuk2L∞ (0,T ;H 1

Γ0 (Ω))

+ kut k2Lm ((0,T )×Γ1 ) .

Then the embedding ZT ,→ C([0, T ]; Lp0 (Γ1 )) is compact. We can now give the Proof of Theorem 1. We set, for any T > 0, the Banach space YT = {u ∈ C([0, T ]; HΓ10 (Ω)), ut ∈ Lm ((0, T ) × Γ1 )}, endowed with the norm kukYT = kukZT , and the closed convex set XT = XT (u0 ) = {u ∈ YT : u(0) = u0 }. Take u ∈ XT . Assumption (5) yields that (p − 1)m0 < r and then, since Γ1 is 0 bounded, using H˝older’s inequality, we have |u|p−2 u ∈ Lm ((0, T ) × Γ1 ). Then, by Theorem 4, there is a unique solution v of the problem  vt − ∆v = 0 in (0, T ) × Ω,     v = 0 on [0, T ) × Γ0 , (45) ∂v m−2 p−2  = −|vt | vt + |u| u on [0, T ) × Γ1 ,     ∂ν v(0, x) = u0 (x) on Ω. We denote by v = Φ(u) the solution v of (45) which corresponds to u, so that Φ : XT → XT by (18) and (19). We apply Schauder fixed point theorem (see [7, Corollary 3.6.2]) to Φ : BR → BR , where BR = {u ∈ XT : kukYT ≤ R}, provided that R is suitably large and T is suitably small. Note that BR is closed and convex, and it is non empty for R ≥ R0 := ku0 kHΓ1 (Ω) . 0 Step 1. We show that Φ maps BR into itself for R sufficiently large and T small enough. Let u ∈ BR . Writing the energy identity (20) for v, with g = |u|p−2 u and s = 0 and using H˝older’s inequality, we obtain, for 0 ≤ t ≤ T , Z t Z t 1 2 2 (46) k∇v(t)k2 + kvt k2 + kvt km m,Γ1 2 0 0 Z TZ 1 ≤ k∇u0 k22 + |u|p−1 |vt | 2 0 Γ1 ¶1/m0 Z T µZ 1 2 (p−1)m0 ≤ k∇u0 k2 + |u| kvt km,Γ1 . 2 0 Γ1 Since (p − 1)m0 < r by (5), applying H˝older inequality in the space variable again, and then the trace–Sobolev embedding and H˝older’s inequality in the time variable, we obtain, for 0 ≤ t ≤ T (here and in the rest of the proof ki will denote positive constants depending only on p, m, Γ1 and Ω), Z t Z t 1 k∇v(t)k22 + kvt k22 + kvt km (47) m,Γ1 2 0 0 Z T 1 2 ≤ k∇u0 k2 + k1 kukp−1 r,Γ1 kvt km,Γ1 2 0 0 1 ≤ k∇u0 k22 + k2 Rp−1 T 1/m kvt kLm ((0,T )×Γ1 ) . 2

13

Writing (47) with t = T , disregarding the first and the second addendum in the left hand side and using Young’s inequality we get 1 2 p−1 1/m0 kvt km T kvt kLm ((0,T )×Γ1 ) Lm ((0,T )×Γ1 ) ≤ k∇u0 k2 + k2 R 2 1 1 (p−1)m0 ≤ k∇u0 k22 + kvt km . Lm ((0,T )×Γ1 ) + k3 T R 2 m Using the inequality (A + B)τ ≤ max{1, 2τ −1 }(Aτ + B τ ),

(48)

for A, B ≥ 0, τ > 0,

we have 2/(m−1)

(49)

kvt kLm ((0,T )×Γ1 ) ≤ k4 (R0

+ R(p−1)/(m−1) T 1/m ).

Using (49) in (47) we obtain, for 0 ≤ t ≤ T , Z t 0 1 1 2/(m−1) (50) k∇v(t)k22 + kvt k22 ≤ R02 + k5 Rp−1 T 1/m (R0 + R(p−1)/(m−1) T 1/m ) 2 2 0 and then, for t = T , 0 1 2/(m−1) (51) kvt k2L2 ((0,T )×Ω) ≤ R02 + k5 Rp−1 T 1/m (R0 + R(p−1)/(m−1) T 1/m ). 2 Using (48) with τ = 2 and H˝older inequality in time we have µ ¶2 Z t 2 kv(t)k2 ≤ ku0 k2 + kvt k2 ≤ 2ku0 k22 + 2T kvt k2L2 ((0,T )×Ω) 0

≤ 2R02 + 2T kvt k2L2 ((0,T )×Ω) , and consequently, using (51), (52)

0

2/(m−1)

kv(t)k22 ≤ (2 + T )R02 + 2k5 Rp−1 T 1+1/m (R0

+ R(p−1)/(m−1) T 1/m ).

Combining (50) and (52) (53)

kvk2L∞ (0,T ;H 1

Γ0 (Ω))

0

≤ (3 + T )R02 + 2k5 Rp−1 T 1/m (1 + T ) 2/(m−1)

×(R0

+ R(p−1)/(m−1) T 1/m ).

Thanks to (49) and (53) to prove that v ∈ BR it is enough to show that (54)

2/(m−1)

k4 (R0

+ R(p−1)/(m−1) T 1/m ≤ R/2

and 0

2/(m−1)

+ R(p−1)/(m−1) T 1/m ) ≤ R2 /4.

2/(m−1)

, 6R0 },

(55) (3 + T )R02 + 2k5 Rp−1 T 1/m (1 + T )(R0 We first fix (56)

R := max{1, 4k4 R0

so that (57)

2/(m−1)

k4 R0

and 3R02 ≤ R2 /8.

≤ R/4

We then take T ≤ 1 small enough so that k4 R(p−1)/(m−1) T 1/m ≤ 1/2, (58)

0

2/(m−1)

4K5 Rp−1 T 1/m R0 0

T R02 ≤ 1/16, ≤ 1/32,

4k5 Rp−1 T 1/m R(p−1)/(m−1) T 1/m ≤ 1/32.

14

ENZO VITILLARO

Solving (58) with respect to T and using (56) one immediately see that T = T (R02 , m, p, Ω, Γ1 ) is decreasing in R0 . Moreover, since by (56) one has R ≥ 1, formulas (57)– (58) immediately yield (54)–(55), proving that with such a choice of R and T one has v ∈ BR . Step 2. We now prove that, with this choice of R and T , the map Φ is continuous on BR and that Φ(BR ) is relatively compact in YT . Let u, u ¯ ∈ BR , and denote v = Φ(u), v¯ = Φ(¯ u), w = v − v¯. Clearly w is a solution of the problem  in [0, T ) × Ω,  wt − ∆w = 0   w = 0 on (0, T ) × Γ0 , (59) ∂w  = −|vt |m−2 vt + |¯ vt |m−2 v¯t + |u|p−2 u − |¯ u|p−2 u ¯ on [0, T ) × Γ1 ,    ∂ν  w(0, x) = 0 on Ω. Since vt , v¯t ∈ Lm ((0, T ) × Γ1 ), we also know that |vt |m−2 vt and |v¯t |m−2 v¯t belong 0 to Lm ((0, T ) × Γ1 ). Moreover, by (5) the functions |u|p−2 u and |¯ u|p−2 u ¯ belong to 0 m L ((0, T ) × Γ1 ). Then we can apply Lemma 1, so that the energy identity (29) becomes Z t Z tZ 1 (60) k∇wk22 + kwt k22 + [|vt |m−2 vt − |v¯t |m−2 v¯t ]wt 2 0 0 Γ1 Z tZ = [|u|p−2 u − |¯ u|p−2 u ¯]wt 0

Γ1

for 0 ≤ t ≤ T . Estimating from above the right hand side and from below the third term in the left hand side of (60) exactly as in [27], starting from formula (72) onward we obtain the estimates (61)

kvt − v¯t k2L2 ((0,T )×Ω) ≤ Kku − u ¯kL∞ (0,T ;Lr0 (Γ1 ))

(62)

k∇v − ∇¯ v k2L∞ (0,T ;L2 (Ω)) ≤ Kku − u ¯kL∞ (0,T ;Lr0 (Γ1 ))

and (63)

kvt − v¯t km ¯kL∞ (0,T ;Lr0 (Γ1 )) , Lm ((0,T )×Γ1 ) ≤ Kku − u

when m ≥ 2, while (64)

0

≤ Kku − u ¯kL∞ (0,T ;Lr0 (Γ1 )) , k|vt |m−2 vt − |¯ vt |m−2 v¯t km Lm0 ((0,T )×Γ1 )

when 1 < m < 2, where K = K(p, m, Γ1 , Ω, T, R) > 0. From v(0) = v¯(0) = u0 we get that, for 0 ≤ t ≤ T , Z T kvt − v¯t k2 ≤ T 1/2 kvt − v¯t kL2 ((0,T )×Ω) ; kv(t) − v¯(t)k2 ≤ 0

hence by (61) we get (65)

kv(t) − v¯(t)k2L∞ (0,T ;L2 (Ω)) ≤ KT ku − u ¯kL∞ ((0,T )×;Lr0 (Γ1 ))) .

Estimates (61)–(65) show that Φ is continuous on BR . To prove that Φ(BR ) is relatively compact in YT , let (uk )k be a sequence in BR . By Lemma 2 the sequence uk is relatively compact and then, up to a subsequence, it is a Cauchy sequence, in C([0, T ]; Lr0 (Γ1 )). Hence, applying (61), (62), (63) and (65) with u = ui , u ¯ = uj , k k i, j ∈ N, it follows that, when m ≥ 2, v = Φ(u ) is a Cauchy sequence in YT .

15

When 1 < m < 2, by (64), |vtk |m−2 vtk is a Cauchy sequence in the space 0 L ((0, T ) × Γ1 ). Then, by [4, Th´eor`eme IV.9], there is χ0 ∈ Lm ((0, T ) × Γ1 ) k m−1 such that (up to a subsequence) |vt | ≤ χ0 on (0, T ) × Γ1 and vtk is a.e. convergent in (0, T ) × Γ1 . Then, by the Lebesgue dominated convergence theorem, vtk is convergent in Lm ((0, T ) × Γ1 ). Then, using (61), (62) and (65), v k is a Cauchy sequence in YT also when 1 < m < 2. Since YT is a complete, this concludes the proof. ¤ m0

Now we can give the Proof of Theorem 2. Since T = T (ku0 k2H 1

Γ0 (Ω)

, m, p, Ω, Γ1 ) is decreasing in the first

variable, we can apply the standard continuation procedure of ordinary differential equations (see also [23]) to conclude that either the solution u is global or there is Tmax < ∞ such that (66)

lim ku(t)k2H 1

Γ0 (Ω)

− t→Tmax

= ∞.

We shall prove that (66) cannot happens. We note that u ∈ W 1,p (0, T ; Lp (Γ1 )), since p ≤ m. Moreover, the potential operator u → kukpp,Γ1 is Frech`et differentiable in Lp (Γ1 ). Then Z d (67) ku(t)kpp,Γ1 = p |u(t)|p−2 u(t)ut (t), dt Γ1 for almost all t ∈ (0, T ), and the energy identity (8) can be written in the equivalent form ¯t Z ¯ t¡ ¢ 1 1 ¯ p 2 (68) k∇uk2 − kukp,Γ1 ¯ + kut k22 + kut km m,Γ1 = 0 ¯ 2 p s s

for 0 ≤ s ≤ t < Tmax . We now introduce the auxiliary functional 1 1 (69) H(t) = k∇uk22 + kukpp,Γ1 . 2 p By (67) – (68)

(70)

Z + 2 |u|p−2 uut H0 (t) = − kut k22 − kut km m,Γ1 Γ1 Z ≤ − kut km + 2 |u|p−2 uut . m,Γ1 Γ1

Repeating verbatim the arguments of [27, Proof of Theorem 2] from formula [27, (88)] onward we then prove the estimate (71)

H0 (t) ≤ C(1 + H(t))

with a suitable C(m, p, Γ1 ) > 0. Then, by Gronwall’s Lemma, we have H ∈ L∞ (0, Tmax ). Then, by (67) (72)

kukp,Γ1 , k∇uk2 ∈ L∞ (0, Tmax ).

Moreover, by (68) and (72), Z t 1 1 1 1 kut k22 ≤ k∇u0 k22 + kukpp,Γ1 ≤ k∇u0 k22 + kukpL∞ (0,Tmax ;Lp (Γ1 )) 2 p 2 p 0

16

ENZO VITILLARO

so that (73)

Z kuk22 ≤ 2ku0 k + 2

t 0

2 kut k22 ≤ 2ku0 k22 + k∇u0 k22 + kukpL∞ (0,Tmax ;Lp (Γ1 )) p

which, together with (72), contradicts (66), concluding the proof.

¤

We can now turn to the proof of Theorem 3. We first give an alternative and more explicit characterization of the set W given in (13) and of the number d defined in (12). Since λn−1 (Γ0 ) > 0, as recalled, Poincar`e inequality holds (see [28]) and we can take k∇uk2 as an equivalent norm in HΓ10 (Ω). Then, using the trace–Sobolev embedding, since p < r, (74)

B∞ :=

kukp,Γ1 < ∞. (Ω), u6≡0 k∇uk2

sup 1 u∈HΓ

0

We set

µ λ∞ =

−p/(p−2) B∞

and

E∞ =

1 1 − 2 p

¶ λ2∞ ,

and W1 = {u0 ∈ HΓ10 (Ω) : k∇u0 k2 < λ∞ , We can then give

J(u0 ) < E∞ }.

Lemma 3. Suppose λn−1 (Γ0 ) > 0, and let d be the number and W the set respectively defined in (12) and (13). Then E∞ = d and W = W1 . The proof of Lemma 3 is a straightforward modification of that of [27, Lemma 3]. It is given, for the reader convenience, in Appendix B. Remark 2. The number d = E∞ above is equal (see [26]) to the Mountain Pass level associated to the elliptic problem   ∆u = 0 in Ω,   u=0 on Γ0 ,    ∂u = |u|p−2 u on Γ1 , ∂ν that is to the number (see for example [3]) inf sup J(γ(t)),

γ∈Λ t∈[0,1]

where Λ = {γ ∈ C([0, 1]; HΓ10 (Ω)) : γ(0) = 0,

J(γ(1)) < 0}.

Proof of Theorem 3. By (68) the function t 7→ J(u(t)) is decreasing, then (75)

J(u(t)) ≤ J(u0 )

for t < Tmax .

As in the proof of Theorem 2, using Theorem 1, it is enough to prove that (66) leads to a contradiction. By Lemma 3 we have (76)

k∇u0 k2 < λ∞ ,

and

J(u0 ) < d = E∞ .

By (74) (77)

J(u(t)) ≥

1 1 p k∇uk22 − B∞ k∇ukp2 := h(k∇uk2 ). 2 p

17

Starting from (76)–(77) and repeating verbatim the arguments of [27, Proof of Theorem 3] (with E(t) = J(u(t))) from [27, (98)] onward we get (78)

k∇u(t)k2 < λ∞

for all t ∈ [0, Tmax ).

Still using Poincar`e inequality this contradicts (66), concluding the proof.

¤

Remark 3. To prove Theorem 3 we used, in an essential way, the statement of Theorem 1, that is local existence for solutions of (2). It is also possible to give a more direct proof of Theorem 3, without using this argument. This alternative proof is simpler that the combination of the proofs of Theorems 1 and 3, so should be considered preferable from an ”abstract” point of view. We refer the interested reader to the proof of Theorem 8 in Section 4 where, handling with more general equations, we shall follow this different approach. In the economy of this section the proof we gave is preferable. 4. More general equations This section is devoted to give the extension of our results to problem (1), where Q and f satisfy some specific assumptions which generalize the specific behavior of |ut |m−2 ut and |u|p−2 u. As for the term Q we suppose that there is Θ > 0 such that (Q1) Q is a Carath´eodory real function defined on (0, Θ) × Γ1 × R such that Q(t, x, 0) = 0 for almost all (t, x) ∈ (0, Θ) × Γ1 , and there are an exponent m > 1 and positive constants c1 , c2 , c3 and c4 , possibly dependent on Θ, such that c1 |v|m−1 ≤ |Q(t, x, v)| ≤ c2 |v|m−1 ,

when |v| ≥ 1,

and c3 |v|m−1 ≤ |Q(t, x, v)| ≤ c4 ,

when |v| ≤ 1

for almost all (t, x) ∈ (0, Θ) × Γ1 and all v ∈ R; (Q2) the function Q(t, x, ·) is increasing for almost all (t, x) ∈ (0, Θ) × Γ1 . In most (but not all) of our results we shall also use a stronger version of (Q2), namely (Q3) there is c5 > 0 such that, when m ≥ 2, (79)

(Q(t, x, v) − Q(t, x, w)) (v − w) ≥ c5 |v − w|m while, when 1 < m < 2,

(80)

¯m 0 ¯ (Q(t, x, v) − Q(t, x, w)) (v − w) ≥ c5 ¯|v|m−2 v − |w|m−2 w¯ for almost all (t, x) ∈ (0, Θ) × Γ1 , all v, w ∈ R.

Remark 4. Assumptions (Q1)–(Q3) are satisfied for all Θ > 0 by any Q = Q(v) such that |Q(v)| 1,1 (81) Q ∈ Wloc (R), Q(0) = 0, lim inf m−1 > 0, v→0 |v| |Q(v)| |Q(v)| (82) lim sup m−1 < ∞, lim inf m−1 > 0, |v|→∞ |v| |v|→∞ |v|

18

ENZO VITILLARO

and (83)

Q0 (v) ≥ q0 |v|m−2

for almost all v ∈ R,

for some positive constant q0 . This assertion can be easily checked using the elementary inequality (84)

(|v1 |`−2 v1 − |v2 |`−2 v2 )(v1 − v2 ) ≥ Const.|v1 − v2 |`

for all ` ≥ 2, v1 , v2 ∈ R, with ` = m when m ≥ 2, and with ` = m0 when 1 < m < 2. Of course the damping model term |ut |m−2 ut considered in previous sections satisfies (81)–(83). Two explicit non power–like examples satisfying these requirements are given by ( |v|m−2 v, for |v| ≥ 1, (85) Q0 (v) = 1<µ<m |v|µ−2 v, for |v| ≤ 1, and by (86)

Q1 (v) = |v|µ−2 v + |v|m−2 v,

1 < µ < m.

A time–dependent example satisfying (Q1)–(Q3) for all Θ > 0 is given by (87)

Q2 (t, v) = β(t)|v|m−2 v,

m > 1,

β ∈ C([0, ∞); R+ ).

Remark 5. Let us note that (Q1)-(Q2) yield that there are positive constants c6 and c7 (possibly dependent on Θ) such that (88)

|Q(t, x, v)| ≤ c6 (1 + |v|m−1 ),

and (89)

Q(t, x, v)v ≥ c7 |v|m

for almost all (t, x) ∈ (0, Θ) × Γ1 and all v ∈ R. Concerning the term f , for the time being we assume only that f is a Caratheodory real function defined on Γ1 × R. We shall give case by case the assumptions we need. 4.1. Forced heat equation. Our first result generalizes Theorem 4 of Section 2 to the problem  in (0, T ) × Ω,  ut − ∆u = 0   u = 0 on [0, T ) × Γ0 , (90) ∂u  = −Q(t, x, ut ) + g(t, x) on [0, T ) × Γ1 ,    ∂ν  u(0, x) = u0 (x) on Ω, where g is a given forcing term acting on Γ1 and T > 0 is fixed. Theorem 5. Suppose that (Q1)–(Q2) hold with Θ = T, and that 0

g ∈ Lm ((0, T ) × Γ1 ). Then, given any initial datum u0 ∈ HΓ10 (Ω) there is a unique weak solution u of (90) on (0, T ) × Ω. Moreover (18) – (19) hold, and u satisfies the energy identity Z tZ Z tZ ¯t Z t 1 ¯ k∇uk22 ¯ + (91) kut k22 + Q(·, ·, ut )ut = gut 2 s s s Γ1 s Γ1

19

for 0 ≤ s ≤ t ≤ T . Remark 6. According with Definition 1 a solution of (90) is a function u ∈ L∞ (0, T ; HΓ10 (Ω)) such that ut ∈ L2 ((0, T ) × Ω) ∩ Lm ((0, T ) × Γ1 , (22) is verified with |ut |m−2 ut replaced by Q(·, ·, ut ), and u(0) = u0 . Sketch of the proof of Theorem 5. Repeat the proof of Theorem 4, using (88), (89) and (Q2), starting from the problem ( R R j = 1, . . . , k, (ukt , wj ) + (∇uk , ∇wj ) + Γ1 Q(·, ·, ukt )wj = Γ1 gwj , (92) uk (0) = u0k , instead that from (31), and correspondingly redefining Z (93) Gk (y) = y + Q(·, ·, Bk (x) · y)Bk (x) dx,

y ∈ Rk ,

Γ1

and (94)

1 Gk (y) = |y|2 + 2

Z e ·, Bk (x) · y) dx, Q(·, Γ1

where Z (95)

y

e x, y) := Q(t,

Q(t, x, s) ds,

y ∈ R.

0

¤ 4.2. Local existence. To generalize Theorem 1 we assume that (F1) f (x, 0) = 0 and there are exponents 1 < q < 2 ≤ p and c8 > 0 such that, for almost all x ∈ Γ1 and all u1 , u2 ∈ R, £ ¤ |f (x, u1 ) − f (x, u2 )| ≤ c8 |u1 − u2 |(1 + |u1 |p−2 + |u2 |p−2 ) + |u1 − u2 |q−1 . Assumption (F1) is clearly satisfied by (96)

f0 (x, u) = γ(x)|u|q−2 u + δ(x)|u|p−2 u,

1 < q < p,

provided that γ, δ ∈ L∞ (Γ1 ). This can be seen using the elementary inequality ¯ ¯ ¡ ¢ ¯|u1 |p−2 u1 − |u2 |p−2 u2 ¯ ≤ Const.|u1 − u2 | |u1 |p−2 + |u2 |p−2 (97) for u1 , u2 ∈ R, p ≥ 2. Moreover (F1) is satisfied by any f = f (u), derivable for large |u|, such that (98)

f ∈ C 0,α (R),

f (0) = 0,

|f 0 | = O(|up−2 ) as |u| → ∞

where 0 < α ≤ 1, p ≥ 2 and C 0,α (R) denotes the space of α–H˝older continuous real functions on R. An explicit non–algebraic example of such a function is given by ( |u|ν1 −2 u, |u| ≥ 1, (99) f1 (u) = ν1 , ν2 > 1, |u|ν2 −2 u, |u| ≤ 1, with α = min{1, ν1 , ν2 } and p = max{2, ν1 , ν2 }.

20

ENZO VITILLARO

Remark 7. We remark that an immediate consequence of (F1) is that |f (x, u)| ≤ c9 (1 + |u|p−1 )

(100)

for almost all x ∈ Γ1 and all u ∈ R, for some constant c9 > 0. We can now state Theorem 6. Suppose that (Q1), (Q3), (F1) hold, together with (5) when n ≥ 3. Then, given any initial datum u0 ∈ HΓ10 (Ω) there is T > 0 and a weak solution u of (1) on (0, T ) × Ω. Moreover (18) – (19) hold, together with the energy identity Z tZ Z tZ ¯t Z t 1 2¯ 2 (101) k∇uk2 ¯ + kut k2 + Q(·, ·, ut )ut = f (·, u)ut 2 s s s Γ1 s Γ1 for 0 ≤ s ≤ t ≤ T . Moreover T = T (ku0 kHΓ1

0

(Ω) , m, p, Ω, Γ1 , Θ, c7 , c9 )

is decreasing in the first variable and increasing in c7 . Sketch of the proof. We repeat the proof of Theorem 1, where v = Φ(u) is now the unique solution of  vt − ∆v = 0 in (0, T ) × Ω,     v = 0 on [0, T ) × Γ0 , (102) ∂v  = −Q(t, x, vt ) + f (x, u) on [0, T ) × Γ1 ,     ∂ν v(0, x) = u0 (x) on Ω, given by Theorem 5. We procede exactly as in the proof of Theorem 1, in Step 1 the estimate (46) being replaced by Z t Z t 1 kvt km kut k22 + c7 k∇v(t)k22 + m,Γ1 2 0 0 Z TZ 1 ≤ k∇u0 k22 + c9 (1 + |u|p−1 )|vt | (103) 2 0 Γ1 ¶1/m0 Z T µZ 1 2 m0 −1 (p−1)m0 c9 1 + |u| kvt km,Γ1 , ≤ k∇u0 k2 + 2 2 0 Γ1 and the estimate (47) by (104)

1 k∇v(t)k22 + 2

Z

t 0

Z kut k22

+ c7 0

≤

t

kvt km m,Γ1

0 1 k∇u0 k22 + k1 (1 + Rp−1 )T 1/m kvt kLm ((0,T )×Γ1 ) , 2

where k1 = k1 (p, m, Γ1 , Ω, c5 , c7 , c9 ) > 0. Taking R ≥ 1 we can estimate 1 + Rp−1 ≤ 2Rp−1 and then complete the first step as in the proof of Theorem 1, where now T = T (R02 , m, p, Ω, Γ1 , Θ, c7 , c9 ) ≤ Θ. In this procedure, in particular, (49) is still obtained, where k4 also dependens on c5 , c7 and c9 . The form of (89) immediately yields that T can be taken increasing in c7 .

21

Repeating Step 2 the energy identity (60) becomes Z t Z tZ 1 kut k22 + [Q(·, ·, vt ) − Q(·, ·, v¯t )]wt k∇wk22 + 2 Γ1 0 0 Z tZ = [f (·, u) − f (·, u ¯)]wt 0

Γ1

for 0 ≤ t ≤ T . Estimating from above the right hand side and from below the third term in the left hand side of the last formule exactly as in [27, Proof of Theorem 6] we get ³ ´ ¯kL∞ (0,T ;Lr0 (Γ1 )) + ku − u ¯kq−1 kvt − v¯t k2L2 ((0,T )×Ω) ≤ K ku − u L∞ (0,T ;Lr0 (Γ1 )) ³ ´ k∇v − ∇¯ v k2L∞ (0,T ;L2 (Ω)) ≤ K ku − u ¯kL∞ (0,T ;Lr0 (Γ1 )) + ku − u ¯kq−1 r ∞ L (0,T ;L 0 (Γ1 )) and

³ ´ q−1 r ∞ ≤ K ku − u ¯ k + ku − u ¯ k kvt − v¯t km m 0 r ∞ L (0,T ;L (Γ1 )) L ((0,T )×Γ1 ) L (0,T ;L 0 (Γ1 ))

when m ≥ 2, while 0

(105) k|vt |m−2 vt − |¯ vt |m−2 v¯t km Lm0 ((0,T )×Γ1 ) ³ ´ ≤ K ku − u ¯kL∞ (0,T ;Lr0 (Γ1 )) + ku − u ¯kq−1 r ∞ 0 L (0,T ;L (Γ1 )) when 1 < m < 2, where K = K(p, m, Γ1 , Ω, T, R, c5 , c7 , c9 ) > 0. Starting from these estimates, which generalize (61)–(64), the proof can be completed exactly as for Theorem 1. ¤ 4.3. Global existence for arbitrary initial data. Theorem 2 is generalized as follows Theorem 7. Suppose that (Q1) and (Q3) hold for all Θ > 0, that f satisfy (F1) and (F2) there are c10 > 0 and κ ≥ 0 such that F (x, u) ≥ c10 |u|p − κ

(106)

for almost all x ∈ Γ1 and all u ∈ R, where Z u F (x, u) = f (x, η)dη, 0

and finally that 2 ≤ p ≤ m,

and

p < r.

Then any solution given by Theorem 6 can be extended to the whole of (0, ∞) × Ω. Remark 8. It is worth noting that f0 defined in (96) satisfies (F1) and (F2), provided that γ, δ ∈ L∞ (Γ1 ) and inf Γ1 δ > 0. The same assumption are verified by any f = f (u), derivable for large u, such that (107)

0,α f ∈ Cloc (R),

f (0) = 0,

f 0 (u) ∼ c˜|u|p−2 as |u| → ∞,

with c˜ > 0, as for example the function f1 defined in (99).

22

ENZO VITILLARO

Sketch of the proof of Theorem 7. At first we apply the standard continuation procedure of ordinary differential equations to conclude that either the solution u is global or there is Tmax < ∞ such that (66) holds. Let Tmax be the right endpoint of the maximal interval of existence. Suppose by contradiction that − − Tmax < ∞ and limt→Tmax ku(t)kHΓ1 (Ω) < ∞. Then there is a sequence Tn → Tmax 0 with ku(Tn )kHΓ1 (Ω) bounded. For all n ∈ N the time–translated damping term 0 Q(t − Tn , ·, ·) satisfies (Q1) and (Q3), with c1 , c3 (and then c7 ) independent on n (it is enough to take ci = ci (Tmax )). Hence Theorem 6 can be applied to the Cauchy problem with initial time Tn , and the length Tn0 of the maximal interval of existence [Tn , Tn + Tn0 ) is independent on n. This leads to a contradiction, since in this way we can continue the solution to the right of Tmax . To prove that (66) cannot occurs we generalize the arguments given in the proof of Theorem 2. The energy identity can now be written as (108)

1 k∇uk22 − 2

¯t Z Z tZ ¯ t ¯ F (·, u)¯ + kut k22 + Q(·, ·, ut )ut = 0 ¯ Γ1 s s Γ1

Z

s

for 0 ≤ s ≤ t < Tmax . We now introduce the modified auxiliary functional Z 1 (109) H(t) = k∇uk22 + F (·, u) + κλn−1 (Γ1 ). 2 Γ1 Clearly Z Z H0 (t) = − kut k22 − Q(·, ·, ut )ut + 2 f (·, u)ut Γ1 Γ1 Z Z ≤− Q(·, ·, ut )ut + 2 f (·, u)ut . Γ1

Γ1

Then, arguing exactly as in the proof of [27, Theorem 7], we prove that H ∈ L∞ (0, Tmax ). The proof can be completed as for Theorem 2. ¤ 4.4. Global existence in the potential well. It would be possible to generalize Theorem 3 to problem (1) using Theorem 6 together with a straightforward generalization of the arguments used in the proof of Theorem 3. This approach would have the disadvantage that the nonlinearities f and Q had to verify assumptions (Q1), (Q3) and (F1) (together with assumption (F3) given later). The proof of this fact is left to the interested reader, since we are giving in the sequel a more general result, using a more direct approach inspired to [18] (see also Remark 3). In order to generalize the potential well arguments we need that Q verifies (Q1)– (Q2) and that f verifies (100) together with the following assumption on its primitive F (see (106)) (F3) there is c11 > 0 such that F (x, u) ≤

c11 p |u| p

for almost all x ∈ Γ1 and all u ∈ R.

23

All examples of damping terms Q given in Remark 4 clearly satisfy (Q1)–(Q2) for all Θ > 0. Moreover a further example is given by any Q = Q(v) such that (110)

Q ∈ C(R),

(111)

lim sup |v|→∞

Q(0) = 0,

|Q(v)| < ∞, |v|m−1

|Q(v)| > 0, |v|m−1 |Q(v)| lim inf m−1 > 0, |v|→∞ |v| lim inf v→0

and (112)

Q is increasing.

Examples of functions Q satisfying (110)–(112) but not (81)–(83) are given by Q3 (v) = |v + sin v|m−2 (v + sin v),

m>1

and

 µ−2  v, |v| Q4 (v) = sign v   (|v| − 1)µ−2 (|v| − 1)sign v,

for |v| ≤ 1, for 1 ≤ |v| ≤ 1, for |v| ≥ 2,

1 < µ < m.

Examples of nonlinearities f satisfying (F3) and (100) are given by f0 defined in (96) provided that γ, δ ∈ L∞ (Γ1 ) and γ ≤ 0 almost everywhere on Γ1 . In particular, when γ = −1 and δ = 1 we can consider f2 (u) = |u|p−2 − |u|q−2 u, 1 < q < p. A further example is given by any f = f (u), f ∈ C(R), |f (u)| = O(|u|p−1 ) as |u| → ∞ and f (u)u ≤ Const.|u|p for small u. In particular f3 (u) = sin(e|u| )|u|p−2 u, p ≥ 2, satisfies (F3) and (100), but not (F1), as can be easily seen. Examples Q3 , Q4 and f3 motivate the different approach we are using to handle with (1). To extend Theorem 3 we first need to suitably modify the definition of the stable set W given in (13). We shall use the characterization of W given in Section 3. With this aim we note that, by (74) and (F3), when λn−1 (Γ0 ) > 0, R F (·, u) c11 p Γ1 (113) K∞ := sup ≤ B . p 1 k∇uk p ∞ u∈HΓ (Ω), u6=0 2 0

We set

µ

1 1 − 2 p = +∞ if K1 ≤ 0. We define

λ∞ = (pK∞ )−1/(p−2) , if K1 > 0, and λ∞ = E∞

E∞ =

W = {u0 ∈ HΓ10 (Ω) : k∇u0 k2 < λ∞ ,

and

¶ λ2∞ ,

J(u0 ) < E∞ }.

Remark 9. Clearly, when f (x, u) = |u|p−2 u, the above definitions of λ∞ , E∞ and W are in agreement with those given in Section 3. Moreover, when f (x, u) = σ(x)|u|p−2 u, σ ∈ L∞ (Γ1 ), σ ≥ 0, it can be shown as in [26] that E∞ is the Mountain Pass level associated to the elliptic problem   in Ω, ∆u = 0  u=0 on Γ0 ,   ∂u p−2  = σ|u| u on Γ1 . ∂ν This would not be true if we had kept the definition of E∞ given in Section 3, which is not optimal in this more general case. The last example is f ≡ 0, so that

24

ENZO VITILLARO

λ∞ = E∞ = +∞, and W = HΓ10 (Ω) as expected, since in this case there are no source terms in the equation. We can now state Theorem 8. Let (Q1) and (Q2) hold for any Θ > 0, and (F3), (100) hold. Suppose that m > 1, 2 ≤ p < r and m > r/(r + 1 − p), and that λn−1 (Γ0 ) > 0. Then, given any initial datum u0 ∈ W , there is a global weak solution of (1) on (0, ∞) × Ω. Moreover u(t) ∈ W for all t ≥ 0, u ∈ C([0, ∞); HΓ10 (Ω)),

ut ∈ L2 ((0, ∞) × Ω),

ut ∈ Lm loc ([0, ∞) × Γ1 ),

and the energy identity (101) holds. Finally, if c1 and c3 can be taken independent on Θ then ut ∈ Lm ((0, ∞) × Γ1 ), and if also c2 and c4 can be taken independent on 0 Θ, then also Q(·, ·, ut ) ∈ Lm ((0, ∞) × Γ1 ). Remark 10. Of course the constants ci , i = 1, . . . , 4 are independent on Θ when Q is time independent. Concerning the only time–dependent example we gave in this section, i.e. Q2 (see (87)), clearly c1 and c3 are independent on Θ if and only if limt→∞ β(t) > 0, while c2 and c4 are independent on Θ if and only if limt→∞ β(t) < ∞. Sketch of the proof of Theorem 8. Let X and (wk )k and u0k be as in the proof of Theorem 4. We apply Faedo–Galerkin procedure, so, fixed k ∈ N, we consider the problem (114) ( R R (ukt , wj ) + (∇uk , ∇wj ) + Γ1 Q(·, ·, ukt )wj = Γ1 f (·, uk )wj , j = 1, . . . , k, uk (0) = u0k . Pk To conclude that (114) has a local solution uk (t) = j=1 ykj (t)wj , for some yjk ∈ W 1,1 (0, tk ), j = 1, . . . , k, and tk > 0, we use the arguments of Theorem 4. In this process (33), (34) and (35) are respectively replaced by ( Gk (yk0 (t)) + Ak yk (t) = Fk (yk (t)), (115) yk (0) = y0k , ( yk0 (t) = G−1 k (F(yk (t)) − Ak yk (t)), (116) yk (0) = y0k , and (117)

|G−1 k (Fk (yk ) − Ak yk )| ≤ |Fk (yk )| + kAk k|yk |.

where Gk is defined in (93) and Z Fk (y) = f (·, Bk (x) · y)Bk (x) dx,

y ∈ Rk

Γ1

is continuous in Rk (use (100) and the fact that wj ∈ Lp (Γ1 ), j = 1, . . . , k). The corresponding energy function and the energy identity associated to (114) respectively are Z 1 F (·, uk (t)) (118) E k (t) = k∇uk (t)k22 − 2 Γ1

25

and

Z E k (t) − E k (s) = −

(119)

s

t

kukt k22 −

Z tZ s

Γ1

Q(·, ·, ukt )ukt ≤ 0

for 0 ≤ s ≤ t < tk . We note that, by integrating (100), we easily see that |F (x, u)| ≤ c12 (1 + |u|p )

(120)

for almost all x ∈ Γ1 , all u ∈ R, and c12 = c12 (p, c9 ) > 0. Arguing exactly as in [27, Proof of Theorem 8] we then prove the following estimates: k∇uk k2 ≤ C0 , kukt kL2 ((0,tk )×Ω) ≤ C0

(121)

kukt kLm ((0,tk )×Γ1 ) ≤ C0 , kQ(·, ·, ukt )kLm0 ((0,tk )×Γ1 ) ≤ C0 ,

¯ with k¯ suffifor some C0 = C0 (u0 , m, p, Γ1 , c6 (T ), c7 (T ), c12 ) > 0 and for k ≥ k, ciently large (here c6 and c7 are the constant appearing in (88) and (89) for Θ = T ). Starting from estimates (121), and using Lemma 2 with p0 = p, a standard ¯ and that, up to a subsequence, argument yields that tk = T , k ≥ k, uk → u

weakly∗ in L∞ (0, T ; HΓ10 (Ω)) and strongly in C([0, T ]; Lp (Γ1 )),

ukt → ut Q(·, ·, ukt ) → χ

weakly in L2 ((0, T ) × Ω) and in Lm ((0, T ) × Γ1 ), 0

weakly in Lm ((0, T ) × Γ1 ).

Arguing as in the proof of Theorem 4, we use Lemma 1 to obtain the energy identity 0 (which is possible since (5) and (100) yield f (·, u) ∈ Lm ((0, T ) × Γ1 )) and (Q2) to prove that χ = Q(·, ·, ut ). This yields the existence of a solution on (0, T ) × Ω, for all T > 0. Since the same argument can be applied for the Cauchy problem with any initial time, the solution can be extended to the whole of (0, ∞) × Ω. Finally, by the energy identity and (F3) we obtain, for all t > 0, Z t Z tZ Z Z 1 kut k22 + Q(·, ·, ut )ut ≤ k∇u0 k22 + F (·, u) − F (·, u0 ) 2 0 0 Γ1 Γ1 Γ1 Z 1 c11 ≤ k∇u0 k22 − F (·, u0 ) + kukpp,Γ1 2 p Γ1 and then, arguing as in the proof of Theorem 3, we prove that kukp,Γ1 ∈ L∞ (0, ∞), so that we have ut ∈ L2 ((0, ∞) × Ω) and Q(·, ·, ut )ut ∈ L1 ((0, ∞) × Γ1 ). When c1 , c3 (and then c7 ) can be taken independent on Θ we then have ut ∈ Lm ((0, ∞)×Γ1 ), and if also c2 , c4 (and then c6 ) can be taken independent on Θ we also obtain that 0 Q(·, ·, ut ) ∈ Lm ((0, ∞) × Γ1 ) completing the proof. ¤ Appendix A. Proof of Lemma 1 Denote H = L2 (Ω),

V = HΓ10 (Ω),

W = Lm (Γ1 ),

X = {u ∈ V : u|Γ1 ∈ W }.

Since V is dense in H, using [25, Theorem 2.1] and (25)–(26), we obtain that (122)

u ∈ Cw ([0, T ]; V ).

26

ENZO VITILLARO

The key point is to show that (29) holds. With this aim, fixed 0 ≤ s ≤ t ≤ T , we set θ0 to be the characteristic function of the interval [s, t]. For small δ > 0 let θ(τ ) = θδ (τ ) be 1 for τ ∈ [s + δ, t − δ], zero for τ 6∈ (s, t), and linear in the intervals [s, s + δ] and [t − s, t]. Next let ηε be a standard mollifying sequence, that R +∞ is η = ηε ∈ C ∞ (R), supp ηε ⊂ (−ε, ε), −∞ ηε = 1, ηε even and nonnegative, and ηε = ε−1 η1 (τ /ε). Let ∗ denote time convolution. We approximate u, extended as zero outside [0, T ], with v = η∗(θu) ∈ Cc∞ (R; V ). Then Z +∞ Z +∞ d (123) 0= k∇vk22 = (∇v, ∇vt ). −∞ dt −∞ Using standard convolution properties and Leibnitz rule we see that vt = η ∗ (θ0 u) + η ∗ (θut ) in H, so that η ∗ (θut ) ∈ Cc∞ (R; V ). Then, by (123), Z +∞ Z +∞ 0 (124) 0= (η ∗ (θ∇u), η ∗ (θ ∇u)) + (η ∗ (θ∇u), ∇(η ∗ (θut ))). −∞

−∞

Using (26) we can take φ = η ∗ η ∗ (θut ) in (27). Then, multiplying by θ, integrating from −∞ to ∞ and using standard properties of convolution, we can evaluate the second term in (124) in the following way: Z +∞ Z +∞ Z (η ∗ (θ∇u), ∇(η ∗ (θut )) = η ∗ (θζ) η ∗ (θut ) −∞ −∞ Γ1 (125) Z +∞ − −∞

kη ∗ (θut )k22 .

Combining (124) and (125), and recalling that θ = θδ , we obtain the first approximate energy identity Z +∞ Z +∞ kη ∗ (θδ ut )k22 0= (η ∗ (θδ ∇u), η ∗ (θδ0 ∇u)) − −∞ −∞ (126) Z +∞ Z + η ∗ (θδ ζ) η ∗ (θδ ut ) := I1 + I2 + I3 . −∞

Γ1

Now we shall examine each term in (126) separately as δ → 0 and ε (i.e. η) is fixed. Since θδ → θ0 a.e.,kη ∗ (θδ ζ)km0 ,Γ1 ≤ kζkm0 ,Γ1 , kη ∗ (θδ ut )km,Γ1 ≤ kut km,Γ1 and kη ∗ (θδ ut )k2 ≤ kut k2 , using (23), (26) and the Lebesgue dominated converge theorem we have Z +∞ I2 → (127) kη ∗ (θ0 ut )k22 , −∞

and Z (128)

+∞

Z

I3 →

η ∗ (θ0 ζ) η ∗ (θ0 ut ). −∞

Γ1

Next we decompose the term I1 as Z +∞ Z +∞ 0 I1 = (η ∗ (θ0 ∇u), η ∗ (θδ ∇u)) + (η ∗ [(θδ − θ0 )∇u], η ∗ (θδ0 ∇u)) (129) −∞ −∞ :=I4 + I5 .

27

Since θδ → θ0 in L1 (R), by (25) we have that η ∗ [(θδ − θ0 )∇u] → 0 strongly in L∞ (0, T ; H). Moreover, by (25), kη ∗ (θδ0 ∇u)kL1 (0,T ;H) ≤kθδ0 kL1 (R) kηkL∞ (R) k∇ukL∞ (0,T ;H) ≤2kηkL∞ (R) k∇ukL∞ (0,T ;H) so that (130)

I5 → 0

as δ → 0.

Next we note that, by the properties of convolution and the specific form of θδ Z +∞ I4 = θδ0 (η ∗ η ∗ (θ0 ∇u), ∇u) 1 = δ

−∞ Z s+δ s

1 (η ∗ η ∗ (θ0 ∇u), ∇u) − δ

Z

t

(η ∗ η ∗ (θ0 ∇u), ∇u). t−δ

By (122) the function (η ∗ η ∗ (θ0 ∇u), ∇u) is continuous so (131)

I4 → (η ∗ η ∗ (θ0 ∇u)(s), ∇u(s)) − (η ∗ η ∗ (θ0 ∇u)(t), ∇u(t))

as δ → 0. Combining the convergences (128)–(131), recalling that η = ηε and letting ρε = ηε ∗ ηε , we obtain the second approximate energy identity Z +∞ Z Z +∞ (132) (ρε ∗ (θ0 ∇u), ∇u) = ηε ∗ (θ0 ζ) ηε ∗ (θ0 ut ) − kηε ∗ (θ0 ut )k22 . −∞

Γ1

−∞

Now we consider the convergence of the two sides of (132) as ε → 0. By standard arguments, using (26), ρε ∗ (θ0 ut ) → θ0 ut strongly in Lm ((0, T ) × Γ1 ) and in L2 ((0, T ) × Ω), so that, using (23), the right hand side of (132) tends to Z +∞ Z Z +∞ Z tZ Z +∞ θ02 ζut − kθ0 ut k22 = ζut − kut k22 . −∞

Γ1

−∞

s

Γ1

−∞

As for the left hand side of (132) we note that supp ρε ⊂ (−2ε, 2ε), 0 ≤ ρε = O(ε−1 ) and Z +∞ Z 0 Z 1 +∞ 1 ρε = ρε = ρε = . 2 2 0 −∞ −∞ Therefore, for sufficiently small ε, Z +∞ 1 (ρε ∗ (θ0 ∇u)(t), ∇u(t)) − k∇u(t)k22 = ρε (τ )(∇u(t − τ ) − ∇u(t), ∇u(t)) dτ. 2 0 Since, by (122), τ 7→ (∇u(t − τ ) − ∇u(t), ∇u(t)) is continuous and is zero for τ = 0, we conclude that, as ε → 0, 1 (ρε ∗ (θ0 ∇u)(t), ∇u(t)) → k∇u(t)k22 . 2 The same result, of course, continues to hold when t is replaced by s. Then we can pass to the limit in (132) and conclude the proof of (29). To show that (28) holds, we note that, by (29) it follows that t 7→ k∇u(t)k22 is continuous. Then, by (25)–(26), also t 7→ ku(t)k2V is continuous. Now fix t in [0, T ] and let tk → t. Using (122) we have ku(tk ) − u(t)k2V = ku(tk )k2V + ku(t)k2V − 2hu(tk ), u(t)iV → 0, as k → ∞, concluding the proof.

28

ENZO VITILLARO

Appendix B. Proof of Lemma 3 An easy calculation shows that, for any u ∈ HΓ10 (Ω) such that u|Γ1 6≡ 0, we have µ ¶µ ¶2p/(p−2) 1 1 k∇uk2 sup J(λu) = J(λ(u)u) = − , 2 p kukp,Γ1 λ>0 where

2/(p−2)

λ(u) =

k∇uk2

p/(p−2)

kukp,Γ1

. −2p/(p−2)

Then, by (74), it is easy to see that d = (1/2 − 1/p)B∞ , so that d = E∞ . To show that W = W1 we first prove that W ⊂ W1 . Let u0 ∈ W . Hence K(u0 ) ≥ 0, and so ku0 kpp,Γ1 ≤ k∇u0 k22 by (10). Moreover J(u0 ) < d = E∞ , then µ ¶ 1 1 E∞ > − k∇u0 k22 . 2 p Now, if k∇u0 k2 ≥ λ∞ , it follows that E∞ > (1/2 − 1/p)λ2∞ = E∞ , a contradiction. In conclusion W ⊂ W1 . To prove that W1 ⊂ W let u0 ∈ W1 , and suppose by contradiction that K(u0 ) < 0. Then, using (74), p k∇u0 kp2 , k∇u0 k22 < ku0 kpp,Γ1 ≤ B∞ −p/(p−2)

and hence k∇u0 k2 > B∞

= λ∞ , a contradiction. References

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