Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

In this paper the long time behaviour of the solutions of 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega) and then it is proved that this global attractor is a bounded subset of H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega).

In [14] the existence of a global attractor for (1.5) with critical source term (i.e. in the case when the growth of f is of order 5) was proved. However, the regularity of the global attractor in that article was established only in the subcritical case. For the critical case, the regularity of the global attractor of (1.5) was proved in [15], under the assumptions f ∈ C 1 (R), |f ′ (s)| ≤ c(1 + |s| 4 ), ∀s ∈ R and lim inf |s|→∞ f ′ (s) > −λ 1 (1. 6) or f ∈ C 2 (R), |f ′′ (s)| ≤ c(1 + |s| 3 ), ∀s ∈ R and lim inf where λ 1 is a first eigenvalue of −∆ with zero Dirichlet data. In that article the authors obtained a regular estimate for w tt (when w(t, x) is a weak solution of (1.5)) and then proved the asymptotic regularity of the solution of the non-autonomous equation −∆w t − ∆w + f (w) = g − w tt . In [5] and [19], the regularity of the global attractor of (1.5) was proved under the following weaker condition on the source term: In [8], the authors investigated the weak attractor for the quasi-linear strongly damped equation under the following conditions on the nonlinear functions f and ϕ: ϕ ∈ C 2 (R 3 , R), a 2 |η| p−1 |ξ| 2 ≤ 3 i,j=1 ∂ 2 ϕ(η) ∂η i ∂η j ξ i ξ j ≤ a 3 (1 + |η| p−1 ) |ξ| 2 , ∀ξ, η ∈ R 3 , for some a i > 0, (i = 1, 2, 3), C > 0, q > 0 and p ∈ [1,5). When ∂ 2 ϕ ∂ηi∂ηj = 0, (i, j = 1, 2, 3), the strong attractor has also been studied. Recently, in [3], the authors have studied the global attractor for the strongly damped abstract equation However, the approaches of the articles mentioned above, in general, do not seem to be applicable to (1.1). The difficulty is caused by the term σ(w)w t , when the function σ(·) is not differentiable and the growth condition imposed on σ(·) is critical. In this paper we prove the existence of the global attractors for (1. . Then using the embedding H 3 2 +ε (Ω) ⊂ C(Ω) we show that these attractors coincide.

Well-posedness and the statement of the main result
We start with the conditions on nonlinear terms f and σ.
By the standard Galerkin's method it is easy to prove the following existence theorem: 3) hold. Then for every T > 0 and every , which satisfies the following energy equality Now using the method of [16, Proposition 2.2] let us prove the following uniqueness theorem: If w(t, ·) and w(t, ·) are the weak solutions of (1.1)-(1.3), determined by Theorem 2.1, with initial data (w 0 , w 1 ) and ( w 0 , w 1 ) respectively, then where c : R + × R + → R + is a nondecreasing function with respect to each variable and R = max { (w 0 , w 1 ) H , ( w 0 , w 1 ) H }.
Proof. By the assumption of the lemma, there exists a subsequence {v n k } such that v n k → v a.e. in Ω. Then by Egorov's theorem, for any ε > 0 there exists a subset A ε ⊂ Ω such that mes(A ε ) < ε and v n k → v uniformly in Ω\A ε . Hence for large enough k |v n k (x)| ≤ 1 + |v(x)| in Ω\A ε and consequently Applying Lebesgue's theorem we get On the other hand since we have is satisfied. The last inequality together with (2.8) implies that Proof. Let (w 0n , w 1n ) → (w 0 , w 1 ) strongly in H. Denoting (w n (t), w tn (t)) = S(t)(w 0n , w 1n ), (w(t), w t (t)) = S(t)(w 0 , w 1 ) and u n (t) = w n (t) − w(t), by (1.1) we have Since, by Theorem 2.1, every term of the above equation belongs to L 2 (0, T ; H −1 (Ω)), testing it by u nt , we obtain Applying Gronwall's lemma we have (Ω)) = 0, which together with (2.9) yields that S(T )(w 0n , w 1n ) → S(T )(w 0 , w 1 ) strongly in H, for every T ≥ 0. Now let us recall the definition of a global attractor. Definition ( [17]). Let {V (t)} t≥0 be a semigroup on a metric space (X, d). A compact set A ⊂ X is called a global attractor for the semigroup Our main result is as follows: then using the methods of [5] , [19] and [21] one can prove Theorem 2.4. If we assume , then the method of [15] can be applied to (1.1)-(1.3). In this case, as in [20], one can show that a global attractor A attracts every bounded subset of H in the topology of H 1 0 (Ω) × H 1 0 (Ω). Remark 2.2. We also note that problem (1.1)-(1.3), in 3-D case, without the strong damping −∆w t was considered in [11] and [16]. In this case, when σ(·) is not globally bounded, the existence of a global attractor in the strong topology of H and the regularity of the weak attractor remain open (see [11] and [16] for details).

Existence of the global attractor in H
We start with the following asymptotic compactness lemma: ) and integrating over (s, T ) × Ω, we obtain where Σ(w) = w 0 sσ(s)ds. Integrating the last inequality with respect to s from 0 to T we find Since for every ε > 0 the embedding Applying Lemma 2.1 it yields that for small enough ε > 0. The last approximation together with (2.3) and (3.2) 2 implies that Also applying Fatou's lemma and using (2.1), for T ≥ 3+2λ1 λ1 . Now let us estimate the right hand side of (3.9). By (2.1), (3.1) 1 and (3.2), we find that

Existence of the global attractor in H 1
To prove the existence of a global attractor in H 1 we need the following lemmas: Proof. We use the formal estimates which can be justified by Galerkin's approximations. Multiplying both sides of (1.1) by −∆w t and integrating over Ω, we obtain d dt (Ω) , ∀t ≥ 0.
, w (m) (t) , ∀t ∈ R. (5.14) Let us estimate each term on the right hand side of (5.14). By the definition of w (m) , we have , ∀t ∈ R.