Strong Exponential Attractors for Weakly Damped Semilinear Wave Equations

In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain of R 3 . The exponential attractor is investigated in a strong energy space for the case of subquintic nonlinearity, which is based on the recent extension of the Strichartz estimate for the case of a bounded domain. The results obtained complete some previous works.


Introduction
Let Ω ⊂ R 3 be a bounded domain with smooth boundary. Given c > 0, we consider the following weakly damped wave equation: x ∈zΩ , t > 0.
Here, g ∈ L 2 (Ω) is independent of time and the nonlinear function f ∈ C 3 (R) with f(0) � 0 and satisfies the following conditions: where 3 ≤ p < 5 and λ 1 > 0 is the first eigenvalue of −Δ on Ω with Dirichlet boundary condition. Denote by 〈·, ·〉 and ‖·‖ the inner product and norm in L 2 (Ω). For the sake of convenience, we define Hilbert spaces H � H 1 0 (Ω) × L 2 (Ω), H 1 � (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω), and H 2 � (H 3 (Ω) ∩ H 1 0 (Ω)) × (H 2 (Ω) ∩ H 1 0 (Ω)). Dissipative wave equations arise as an evolutionary mathematical model in various systems for the relevant physical application including electrodynamics, quantum mechanics, and nonlinear elasticity (see e.g., [1]). e longtime behavior of problem (1) is of a big permanent interest (see e.g. [2][3][4][5]), which depends strongly on the growth rate of the nonlinearity f. For a long time, in a bounded smooth domain of R 3 , the global well-posedness for problem (1) holds only in the case of subcubic or cubic nonlinearity (that is, the exponent p in 1.2 less or equals to 3), in which the uniqueness is verified by the technology of Sobolev embedding in general. erefore, the cubic growth rate of nonlinearity had been considered as a critical one for the case of 3-D bounded domain. Consequently, the existence of global attractors for the weakly damped wave equations in the natural energy space H as well as its regularities only had been known for the case p ≤ 3, that is, the nonlinear term f at most can be cubic growth in 3-D bounded domain (see [2,3,5,6] and the references therein). e case of supercubic growth rate is a bit more complicated since the uniqueness of energy weak solutions is unknown for the supercubic case (e.g., see [7]). However, this problem can be overcome using slightly more regular solutions than the energy ones.
ese solutions are the so-called Shatah-Struwe solutions. Recently, with the help of the suitable versions of Strichartz estimates, the global well-posedness of quintic wave equation in terms of Shatah-Struwe solutions in 3-D smooth bounded domains has been obtained in [8][9][10]. Based on the fact that the Strichartz estimates can be generalized to weakly damped wave equation, Kalantarov et al. [11] obtained the well-posedness of weakly damped wave equation in the case of quintic nonlinearity in bounded domain of R 3 , where the solution is the weak solution with extra regularity named Shatah-Struwe solution; moreover, they also considered the global attractor for the case of quintic nonlinearity.
Considering the strong solution, in 1983, Babin and Vishik [3] investigated the damped subcubic wave equation in H. For seeking a compact absorbing set in H, they requested the initial data belonging to H 1 , and they proved the existence of the maximal (H 1 , H) attractor for problem (1). en, Ladyzhenskaya [12] obtained the H 1 attractor for cubic nonlinearity. In [13], the second author verified the H 1 attractor via a different approach. Recently, the authors [14] have obtained the strong global attractor for subquintic nonlinearity.
As mentioned by many authors, the global attractor, however, does not provide an actual control of the convergence rate of trajectories and it can be sensitive to perturbations. In order to overcome these difficulties, Eden et al. introduced the notion of exponential attractor (see [15,16]). In contrary to the global attractor, the exponential attractor is not unique, and it is only semiinvariant. However, it has the advantage of being stable with respect to perturbations, attracts the trajectories exponentially, and has finite fractal dimension. For the strongly damped wave equation, in the subcritical case, Pata and Squassina [17] established an exponential attractor of optimal regularity by using the bootstrap argument; in the critical case, Sun and Yang [18,19] obtained the exponential attractors based on some asymptotic regularity of the solution. For the weakly damped wave equation, Eden et al. [20] and Grasselli and Pata [21] investigated the exponential attractor for problem (1) with cubic nonlinearity in H 1 and H, respectively. For the supercubic nonlinearity, in [22], we established the exponential attractor in H. However, to the best of our knowledge, there is no result on exponential attractor for problem (1) in H 1 .
In the present paper, as a continuation of our previous works [14,22], we concentrate on the existence of exponential attractor in the subquintic but supercubic case and establish an exponential attractor in the strong topology space based on the Shatah-Struwe solution.
Our main result can be stated as follows.
(Ω) and f satisfy (2). en, the solution semigroup S(t) acting on H 1 possesses an exponential attractor A 1 which is compact in H 1 , satisfying the following conditions: (iii) ere exist a monotone increasing function Q 1 (·) and positive constant μ such that for any bounded set e above theorem will be proved in Section 3. Before that, the preliminary things, including some notations, the well-posedness, and the global attractor of the system are discussed in Section 2.
roughout the rest of the paper, denote by C any positive constants which may be different from line to line even in the same line; we also denote the different positive constants by C i , i ∈ N, for special differentiation.

Well-Posedness and Global Attractor
Firstly, we recall the definitions about the weak solution and strong solution of problem (1) in [11], which will be used to state our results.
Definition 1 (see [11]). For any T > 0, a function u(t) is a weak solution of problem (1) if (u, u t ) ∈ L ∞ (0, T; H) and equation (1) is satisfied in the sense of distribution, i.e., For the well-posedness of strong solution, we have the following result.
Theorem 2 (see [14]). Let g ∈ L 2 (Ω) and f satisfy (2). en, for every (u 0 , u 1 ) ∈ H 1 , there exists a unique global strong solution u(t) of problem (1) with energy estimate where monotone function Q is independent of u and t. Moreover, for any two strong solutions u 1 (t) and u 2 (t), there exists a positive constant C 1 depending on the H 1 -norm of u 1 (t) and u 2 (t) such that where u(t) � u 1 (t) − u 2 (t). erefore, the strong solution generates a C 0 operator semigroup S(t) { } t≥0 in H 1 as follows: which is continuous in H 1 .
Finally, we state the existence of global attractor in strong energy space which will be used in Lemma 3.

Proof of Theorem 1
ere are two different approaches to find exponential attractors. e original one (cf. [15,16]) relies on the proof that the semigroup S(t) satisfies squeezing property, and the following approach (cf. [16,17,23]) is established via decomposition of semigroup, which is easier to verify for damped wave equation. Lemma 1. Let X 0 and X 1 be two Banach spaces such that X 1 is compactly embedded into X 0 and S(t) { } t≥0 is a semigroup on X 0 . Assume that there exists a bounded positively invariant X ⊂ X 0 and a time t * > 0 such that the following hold: is Lipschitz continuous (with the metric inherited from X 0 ). (ii) e map S(t * ): X ⟶ X admits a decomposition of the form where S 0 and S 1 satisfy the conditions for some C * > 0.
en, the local dynamical system (X, S(t)) possesses an exponential attractor.
In order to verify the local exponential attractor in X is an exponential attractor in H 1 , the following transitivity of exponential attraction is necessary.
We will verify all conditions of Lemma 1 step by step.
Step 1. Construct a bounded positively invariant set. Decompose the solution u(t) as where v(t) solves the linear problem and the remainder w(t) satisfies It is convenient to denote where z � (u 0 , u 1 ). For equation (15), it is easy to check that the solution is exponentially decaying, namely, holds for some ρ > 0. From [14,Lemma 4.2], we have the following regularity estimate for S w (t): with α > 0 small enough.
Proof. After vanishing the v-component, we can obtain a "perturbed positive trajectory." And the original is asymptotically close to positive trajectory exponentially. erefore, a collection of perturbed positive trajectories will be a promising candidate for an exponentially attracting set. To carry out this idea, denote dist H 1 (z, A) < 1} is a neighborhood of the global attractor A. We will show that K 0 is an exponentially attracting set.

Journal of Mathematics
Notice that N is an absorbing set. Hence, for any given bounded set B ⊂ H 1 , there exists T � T(‖B‖ H 1 ) such that S(t)B ⊂ N, for all t ≥ T. Furthermore, for t ≥ T, we have On the other hand, by inequality (5), we have the uniform estimate for some C 5 depending only on ‖B‖ H 1 . Combining the two above inequalities, we obtain dist H 1 S(t)B, K 0 ≤ C 5 e ρT + C 3 e ρT ‖N‖ H 1 +‖g‖ e − ρt , en, we define a positive invariant set by K 1 � ∪ t≥0 S(t)K 0 V , which satisfies dist H 1 S(t)B, K 1 ≤ C 5 + C 3 e ρT ‖N‖ H 1 +‖g‖ e − ρt , t ≥ 0.