Global Attractors for a Class of Weakly Damped Wave Equations with Gradient Type Nonlinearity

In this paper, the main purpose is to study existence of the global attractor for the weakly damped wave equation with gradient type nonlinearity. To this end, we first verify the existence and uniqueness of global weak solution by the Galerkin method and compulsively variational method. Furthermore, we obtained the global strong solution under some mild assumptions on f. Secondly, we utilize the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document}-limit compactness to show the semigroup generated by the equation has a compact, connected and invariant attractor.


Introduction
In this paper we are concerned with the following weakly damped wave equation with gradient type nonlinearity on a bounded domain Ω ⊂ ℝ n with smooth boundary Ω where > 0, p > 1, and 1 < p < n n−2 , n ≥ 3;1 < p < ∞, n = 1, 2, respectively. The source f ∈ C 1 and satisfies the following growth condition where g ∈ L 2 (Ω × (0, T)), q ≤ p+1 2 , s ≤ 1 for some constant C 1 , C 2 > 0. Similar initial-boundary problem was investigated by Meng and Zhu in [18,28] for the equation where > 0, is the nonlinear term, and f is a given external forcing term, Ω ⊂ ℝ 3 is a bounded domain with smooth boundary Ω . Nonlinear wave equation of the type (1.3) arises as an evolutionary mathematical model in many branched of physics, for example, (i) modeling a continuous Josephson junction with (u) = sin u ; (ii) modeling a relativistic quantum mechanics with (u) = |u| 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(see,e.g. [14,21] and references therein). One way to treat this problem is to analyse the existence of its global attractor. Meng obtained the existence of a global attractor for (1.3) in strong topological space H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω) where the nonlinear term with some polynomial growth and the external forcing term f is independent of time. Zhu invested the case which (u) = 0 with nonlinear boundary conditions. The boundary was divided into two parts: one part is Dirichlet boundary and the other part is with boundary damping which the framework was arising in control theory in [12,13]. And also, the other way to treat this problem is to concern the decay of solutions (see,e.g. [4,19,23] and references therein).
Attractor is an important concept describing asymptotic properties of dynamical systems, a great deal of work has been devoted to the existence of global attractors of dynamical systems (see, e.g. [1, 3, 5, 6, 8-10, 22, 26, 27] and references therein). The existence of a global attractor Eq. (1.1) which the source term only containing f was proved by Hale [8] for f satisfying for n ≥ 3 the growth condition f (u) ≤ C 0 (|u| + 1) , with 1 ≤ < n n−2 . For the case n = 2 , Hale and Raugel [7] proved the existence of the attractor under an exponential growth condition of the type |f (u)| ≤ exp (u) (such a condition previously appearing in the work of Gallouë t [16]). The existence of the attractor in the critical case = n n−2 was first proved by Babin and Vishik [2], and then more generally by Arrieta, Carvalho and Hale [5]. For other treatments see Chepyzhov and Vishik [5], Ladyzhenskaya [11], Raugel in Ω, [24] and Temam [26]. When Ω is bounded and u is subjected to suitable boundary conditions, the general result is that the dynamical system associated with the problem possesses a global attractor in natural energy space H 1 0 (Ω) × L 2 (Ω) if nonlinear term f has subcritical or critical exponent, because there exists typical parabolic-like flows with an inherent smoothing mechanism. However, the nonlinear source f with gradient in Eq. (1.1) which is different from [3,6,27,28]. So we make the nonlinear source f those under some growth conditions controlled by the variational structure. By the traditional method [25] (for examples), in order to obtain the existence of global attractors for semilinear wave equations, one needs to verify the uniform compactness of the semigroup by getting the boundedness in a more regular function space. However, in some cases it is difficult to obtain the uniform compactness of the semigroup. Fortunately, a new method for obtaining the global attractors has been developed in [17]. With this method, one only needs to verify a necessary compactness condition( -limit compactness) with the same type of energy estimates as those for establishing the absorbing sets. In this paper, we use this method to obtain the existence of global attractors for the Eq. (1.1) with the complicated condition where the source term f (u, ∇u, x, t) is without variational structure.
This paper is organized as follows: -in Sect. 2 we obtained the existence and uniqueness of weak solution by using Galerkin method and compulsively variational method; -in Sect. 3 we obtained the existence of global attractor for the Eq. (1.1) by using the new method ( -compactness condition).

1) has a global weak solution
Proof Fix spaces as follows: ).

3 3 Existence of Global Attractor
Next, we introduce the concepts and definitions of invariant sets, global attractors, and -limit compactness sets for the semigroup S(t).
Definition 3.1 Let S(t) be a semigroup defined on X. One set Σ ⊂ X is called an invariant set of S(t) if S(t)Σ = Σ, ∀t ≥ 0 . An invariant set Σ is an attractor of S(t) if Σ is compact, and there exists a neighborhood U ⊂ X of Σ such that for any u 0 ∈ U, In this case, we say that Σ attracts U. Especially, if Σ attracts any bounded set of X, Σ is called a global attractor of S(t) in X.

Definition 3.2 Let X be an infinite dimensional Banach space and A be a bounded subset of X. The measure of noncompactness (A) of A is defined by
Lemma 3.1 [16] If A n ⊂ X is a sequence bounded and closed sets, A n ≠ �, A n+1 ⊂ A n , and (A n ) → 0, (n → ∞) , then the set A = ∩ ∞ n=1 A n is a nonempty compact set.

Definition 3.3 [17]
A semigroup S(t) ∶ X → X(t ≥ 0) in X is called -limit compact, if for any bounded set B ⊂ X and ∀ > 0 , there exists t 0 such that where is noncompact measure in X.
For a set D ⊂ X , we define the -limit set of D as follows: where the closure is taken in the X-norm.

Lemma 3.2 [15] Let S(t) be a semigroup in X, then S(t) has a global attractor A in X if and only if
(1) S(t) is -limit compactness, and (2) there is a bounded absorbing set B ⊂ X.
In addition, the -limit set of B is the attractor A = (B).

Remark 3.1
Although the lemma has been proved partly in [15], we still give a proof here. Our proof is different from that in [27] but is similar to that in [17]. We adopt and present the proof also because we will use the same method to obtain the existence of the global attractor.

Proof
Step1. To prove the sufficiency of Lemma 3.2: (a). Since S(t) is -limit compactness, i.e., for any bounded set B ⊂ X and ∀ > 0, there exists a t 0 , such that: and it implies that In conclusion, combine (a-c) and condition (1.2), step 1 has been proved.
Step 2. To prove the necessary of Lemma 3.2.
If A is a global attractor, then the -neighborhood U (A) ⊂ X is a absorbing set. So we need only to prove S(t) is -limit compactness.
Since U (A) is a absorbing set, for any bounded set B ⊂ X and > 0, there exists a time t (B) > 0 such that On the other hand, A is a compact set, there exists finite element x 1 , x 2 , ⋯ , x n ∈ X such that Then, it implies that Hence, the Lemma 3.2. has been proved.

Remark 3.2
Comparing the Remark 3.1., we divide the operator G(u) of Definition (2.1)into two parts: L and T, where L is a linear operator while T is a nonlinear operator. We obtain the global attractor of the problem (1.1) by using the Lemma 2.4.
Proof According to Lemma 3.5., we prove Theorem 3.1 in the following three steps.
Step 1. The problem (1.1) has a globally unique weak solution; Step 2. To prove S(t) has a bounded absorbing set in H 1 0 (Ω) × L 2 (Ω); from the growth condition (1.2), we have and since so we have (3.6) (3.8) combine (3.7), (3.8)and (3.9), we make (3.6) be as follows so where C > 0 . We can conclude that as follows from Gronwall inequality it implies that S(t) has a bounded absorbing set in H 1 0 (Ω) × L 2 (Ω). At last, we prove S(t) is -limit compactness. From the formula in Lemma 2.2, the solution of problem (1.1) can be expressed as follows: Since the linear operator is symmetrical sector operator, it has eigenvalue sequence: (3.10)

Conflict of Interest
The authors have no competing interests to declare that are relevant to the content of this article.

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