Strong Global Attractors for 3D Wave Equations with Weakly Damping

and Applied Analysis 3 |Δu| Ω |Δu x |2dx , respectively. The norms in L Ω , 1 ≤ p < ∞ are denoted by |u|p ∫ Ω |u|pdx , the scalar products of V,H are denoted by


Introduction
Let Ω ⊂ R 3 be a bounded domain with smooth boundary ∂Ω.We consider the following weakly damped wave equation: with the boundary condition u| ∂Ω 0, 1.2 and initial conditions: where α > 0, ϕ is the nonlinear term, and f is a given external forcing term.
Nonlinear wave equation of the type 1.1 arises as an evolutionary mathematical model in many branched of physics, for example, i modeling a continuous Josephson junction with ϕ u β sin u; ii modeling a relativistic quantum mechanics with ϕ u |u| γ u.A relevant problem is to investigate the asymptotic dynamical behavior of these mathematical models.The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics.One way to treat this problem is to analyse the existence of its global attractor.
The existence of global attractors for the classical wave equations in H 1 0 Ω ×L 2 Ω and the regularities of the global attractors has been studied extensively in many monographs and lectures, for example, see 1-7 and references therein.
However, to our knowledge, the research about the stronger attraction of global attractors for the damped wave equations with respect to the norm of 0 Ω is fewer, only has been found in 8-10 .In the above three papers, the global attractors in strong topological space H 2 Ω ∩ H 1 0 Ω × H 1 0 Ω were established, the attraction with respect to the norm of H 2 Ω ∩ H 1 0 Ω × H 1 0 Ω was proved by the asymptotic compactness of the operator semigroup.
Recently, we consider 1.1 in n dimensional space where the nonlinear term ϕ without polynomial growth is in 11 .
In this paper, our aim is to prove the existence of a global attractor for 1.1 in strong topological space where the nonlinear term ϕ with some polynomial growth.For simplicity, we consider the space dimension is 3, as we know, when the space dimension is lagerer than 3, the case is similar as in 3D, when the space dimension is 1 or 2, the case is more easier.The attraction with respect to the norm of Ω will be proved by a method different from 8-10 .Furthermore, this attractor coincides with the global attractor in the weak energy space H 1 0 Ω × L 2 Ω .The basic assumptions about the external forcing term f and the nonlinear term ϕ are as follows.Let f ∈ L 2 Ω be independent of time, and let the nonlinear term ϕ ∈ C 1 R, R satisfy the following assumptions: , endowed with the standard product norms: Denote by C any positive constant which may be different from line to line and even in the same line, we also denote the different positive constants by C i , i ∈ N, for special differentiation.
The rest of the paper is organized as follows.In the next section, for the convenience of the reader, we recall some basic concepts about the global attractors and recapitulate some abstract results.In Section 3, we present our main results.

Preliminaries
In this section, we first recall some basic concepts and theorems, which are important for getting our main results.We refer to 2, 5, 6, 12, 13 and the references therein for more details.Then, we outline some known results about 1.1 -1.3 .
Definition 2.1.The mappings S t , where S : X × 0, ∞ → X, is said to be a C 0 semigroup on X, if {S t } t≥0 satisfies 1 S 0 u u for all u ∈ X; In 12 , the authors have discussed the relations between Condition C and ω-limit compact and proved that, in uniformly convex Banach space, Condition C is equivalent to ω-limit compact, if the semigroup has a bounded absorbing set.
Next, we recall the result about the global attractor in H 0 whose proofs are omitted here, the reader is referred to 6 and the reference therein.

Main Results
According to the standard Fatou-Galerkin method, it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value of 1.1 -1.3 .We address the reader to 6 and the reference therein.Here, we only state the result as follows.

3.3
We define the mappings: By Lemma 3.1, it is easy to see that {S t } t≥0 is C 0 semigroup in the energy phase spaces H 0 and H 1 .
In order to verify the existence of the bounded absorbing set in H 1 , we need the result about the existence of the bounded absorbing set in H 0 .First, we establish the bounded absorbing set in H 0 .Its proof is essentially established in 6 and the reference therein, and we only need to make a few minor changes for our problem.Here, we only give the following lemma.
Lemma 3.2.Under the conditions 1.4 , 1.5 , 1.6 , {S t } t≥0 has a bounded absorbing set B 0 B H 0 0, ρ 0 in H 0 , that is, for any ε > 0 and any bounded subset B 0 ⊂ H 0 , there is a positive constant t 0 t B 0 , ρ 0 such that S t B ⊂ B 0 for any t ≥ t 0 , u 0 , u 1 ∈ B 0 .

6
Abstract and Applied Analysis It follows from 1.5 that, for any ε > 0, there exists a constant C 1 > 0, such that

3.10
where C 2 is the positive constant satisfying

3.12
If u 0 , v 0 belongs to a bounded set B of H 1 , then B is also bounded in H 0 , and for t ≥ t 0 , by Lemma 3.2, we have

3.15
Combining with 3.8 , 3.12 , and 3.15 , by the Hölder inequality and the Young inequality, we deduce from that 3.7 :

and we conclude that
The ball of H 1 , B B H 1 0, ρ 1 , centered at A −1 f, 0 of radius ρ 1 > ρ 1 ρ 0 , is absorbing in H 1 for the semigroup S t , t ≥ 0.
We now give the property of compactness about the nonlinear operator ϕ which will be needed in the proof of the condition C .
Proof.Let {u m } be a bounded sequence in D A .Without loss of generality, we assume that {u m } weakly converges to u 0 in D A , since D A is reflexive.By the Sobolev embedding theorem, we know that Hence, we have that

3.22
Furthermore, there exists a constant C such that

3.23
It is sufficient to prove that {ϕ u m } converges to {ϕ u 0 } in V :

3.24
On the one hand, for the first term in 3.24 , combining with 3.23 and the continuity of ϕ • , we have

3.25
On the other hand, for the second term in 3.24 , using the continuity of ϕ • , follows immediately by dominated convergence theorem.
Also, considering 3.22 , passing to the limit in 3.24 , we can obtain lim m → ∞ ϕ u m − ϕ u 0 0.

3.27
This completes the proof.Proof.Let {ω i } be an orthonormal basis of L 2 Ω which consists of eigenvalues of A. The corresponding eigenvalues are denoted by {λ j } ∞ j 1 : with Aω i λ i ω i , ∀i ∈ N.

3.29
Let V m span{ω 1 , . . ., ω m } in V and let P m : V → V m be an orthogonal projector.We write u P m u I − P m u u 1 u 2 .

3.30
Taking the scalar product of 1.1 in H with Av 2 Au 2t σAu 2 , we find 1 2

3.32
Since f ∈ L 2 Ω , ϕ : D A → V is compact by Lemma 3.4, for any ε > 0, there exists some m such that I − P m f H ≤ ε, 3.33

Theorem 2 . 6 .
Under the conditions 1.4 , 1.5 , 1.6 , the solution semigroup {S t } t≥0 of the problem 1.1 -1.3 has a global attractor A 0 in H 0 .A 0 is included and bounded in H 1 .
| 2 dx 1/2 , respectively.The norms in L p Ω , 1 ≤ p < ∞ are denoted by |u| p Ω |u| p dx 1/p , the scalar products of V, H are denoted by , H * and V * are the dual spaces of H and V , respectively, and each space is dense in the following one and the injections are continuous.Then, we introduce the product Hilbert spaces * ⊂ V * Definition 2.3.A C 0 semigroup {S t } t≥0 in a Banach space X is said to satisfy the condition C if for any ε > 0 and for any bounded set B of X, there exist t B > 0 and a finite dimensional subspace X 1 of X such that { PS t x X , x ∈ B, t ≥ t B } is bounded and Let {S t } t≥0 be a semigroup on a metric space E, d .A set B 0 ⊂ E is called an absorbing set for the semigroup {S t } t≥0 , if and only if for every bounded set B ⊂ E, there exists a T 0 T 0 B > 0 such that S t B ⊂ B 0 for all t ≥ T 0 .
is called a global attractor for the semigroup, if A is compact and enjoys the following properties:1 A is invariant, that is, S t A A, for all t ≥ 0;2 A attracts all bounded sets of E. That is, for any bounded subset B of E, Theorem 2.5.Let X be a Banach space and let {S t } t≥0 be a C 0 semigroup in X.Then, there is a global attractor for {S t } t≥0 in X if the following conditions hold true:1 {S t } t≥0 satisfies the condition C , and 2 there is a bounded absorbing set B ⊂ X.