The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a nonlinear viscoelastic wave equation with strong damping, linear damping and source terms. Then we study the global attractors of the equation.


Introduction
We know that viscoelastic materials have memory effects.These properties are due to the mechanical response influenced by the history of the materials.As these materials have a wide application in the natural science, their dynamics are of great importance and interest.The memory effects can be modeled by a partial differential equation.In recent years, the behaviors of solutions for the PDE system have been studied extensively, and many achievements have been obtained.Many authors have focused on the problem of existence, decay and blow-up for the last two decades, see [1]- [5].And the attractors are still important contents that are studied.
In [6] and they proved the global existence, uniqueness and exponential stability of solutions and existence of the global attractor.In [7], Y.M. Qin, B.W. Feng and M. Zhang considered the following initial-boundary value problem: where Ω is a bounded domain of ( ) with a smooth boundary ∂Ω , ( ) , u x t τ (the past history of u) is a given datum which has to be known for all t τ ≤ , the function g represents the kernel of a memory, ( ) = is a non-autonomous term, called a symbol, and ρ is a real number such that . They proved the existence of uniform attractors for a non-autonomous viscoelastic equation with a past history.For more related results, we refer the reader to [8]- [14].
In this work, we intend to study the following initial-boundary problem: , where 1 2 3 , , 0, ε ε ε ≥ and ( ) , for the problem (1.3), the memory term , and we consider the strong damping term Thus, the original memory term can be written as and we get a new system with the initial conditions , 0 , , 0 , 0 , 0 0, and the boundary conditions The rest of this paper is organized as follows.In Section 2, we first obtain the priori estimates, then in Section 3, we prove the existence of the global attractors.
For convenience, we denote the norm and scalar product in ( )

The Priori Estimates of Solution of Equation
In this section, we present some materials needed in the proof of our results, state a global existence result, and prove our main result.For this reason, we assume that (G1) : Lemma 1. Assume (G1), (G2) and (G3) hold, let ( ) ( ) ( ) here ( ) ( ) , thus there exists 0 E and ( ) Proof.We multiply with both sides of equation and obtain ) For the first term on the right side (2.5), by using (G1), (G2) and (G3), we have where For the second term on the right side (2.5), by using Holder inequality and Young's inequality, we get So, we have By using Poincare inequality, we obtain and ( ) ( ) By using Holder inequality and Young's inequality, we obtain ( ) ( ) Then, we have ( ) Next, we take proper where ( ) ( ) , by using Gronwall inequality, we obtain ( ) ( ) ( ) , according to Embedding Theorem then ( ) ( ) So, there exists 0 E and ( ) Here ( ) ( ) , thus there exists 1 E and ( ) , Proof.We multiply  ( ) By using Holder inequality, Young's inequality and Poincare inequality, we get For the first term on the right side (2.23), by using (G1), (G2) and (G3), we have where For the second term on the right side (2.23), by using Holder inequality and Young's inequality, we get By using Poincare inequality, we have , , (2.28) And using Interpolation Theorem, we have , , , , .22 By using Holder inequality and Young's inequality, we have ( ) ( ) Then, we have Next, we take proper where ( ) ( ) , by Gronwall inequality, we have , according to Embedding Theorem, then , .

Global Attractors
Theorem 1. Assume (G1), (G2) and (G3) hold, let By the method of Galerkin and Lemma 1 and Lemma 2, we can easily obtain the existence of solutions.Next, we prove the uniqueness of solutions in detail.
Assume that , u v are two solutions of equation, let w u v = − , then, the two equations subtract and obtain where By multiplying the equation by w′ and integrating over Ω , we get by using (G1), (G2) and (G3), we have By using Poincare inequality, we have ( ) and ( ) By using Holder inequality, Young's inequality and Poincare inequality, we have By using Gronwall inequality, we have So we get ( ) 0 w t ≡ , the uniqueness is proved.Theorem 2. Let X be a Banach space, and = , here I is a unit operator.Set ( ) S t satisfy the follow conditions.

Lemma 1 to Lemma 2, we can get that
Theorem 3.Under the assume of Theorem 1, equations have global attractor Under the conditions of Theorem 1, it exists the solution semigroup ( ) t exist a compact global attractor A. The proof is completed.
S t is a completely continuous operator A.Therefore, the semigroup operators ( ) S t exist a compact global attractor.S