Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model

In this paper, we study the long time behavior of solution to the initial boundary value problem for a class of Kirchhoff-Boussinesq model flow ( ) ( ) ( ) ( ) ∆ tt t t u u u u div g u u h u f x α β 2 2 + − ∆ + ∆ = ∇ ∇ + + . We first prove the wellness of the solutions. Then we establish the existence of global attractor.


Introduction
In this paper, we are concerned with the existence of global attractor for the following nonlinear plate equation referred to as Kirchhoff-Boussinesq model: , 0 ; , 0 , with clamped boundary condition Ω ⊂  where v is the unit outward normal on ∂Ω .Here 0 k > is the damping parameter, the mapping 2 2 0 : f →   and the smooth functions 1 f and 2 f represent (nonlinear) feedback forces acting upon the plate, in particular, ( ) ( ) ( ) 1 0 f u = , also considering the (1.4) with a strong damping, then (1.4) becomes a class of Krichhoff models arising in elastoplastic flow, which Yang Zhijian and Jin Baoxia [2] studied.In this model, Yang Zhijian and Jin Baoxia gained that under rather mild conditions, the dynamical system associated with above-mentioned IBVP possesses in different phase spaces a global attractor associated with problem (1.6), (1.2) and (1.3) provided that g and h satisfy the nonexplosion condition, ( )

and 1 2
h h h = + and there exist constant ( ) ( ) Zhijian Yang, Na Feng and Ro Fu Ma [3] also studied the global attractor for the generalized double dispersion equation arising in elastic waveguide model In this model, g satisfies the nonexplosion condition, ( ) ( ) ( ) where ( ) T. F. Ma and M. L. Pelicer [4] studied the existence of a finite-dimensional global attractor to the following system with a weak damping.
with simply supported boundary condition where ( ) ( ) For more related results we refer the reader to [5]- [8].Many scholars assume ( ) ( ) to make these equations more normal; we try to make a different hypothesis (specified Section 2), by combining the idea of Liang Guo, Zhaoqin Yuan, Guoguang Lin [9], and in these assumptions, we get the uniqueness of solutions, then we study the global attractors of the equation.

Preliminaries
For brevity, we use the follow abbreviation: , , , , , ⋅ ⋅ for the H-inner product will also be used for the notation of duality pairing between dual spaces.
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 (H 1 ) ( ) where ( ) ( ) , 0 2 ρ < < , and when 2 N ≥ , ( ) ( ) is the first eigenvalue of the −∆ .Now, we can do priori estimates for Equation (1.1).Lemma 1. Assume (H 1 ), (H 2 ) hold, and ( ) where , thus there exists 0 E and ( ) Remark 1. (2.1) and (2.1) imply that there exist positive constants C η and C η  , such that Taking H-inner product by v in (2.7), we have , .
, by using Holder inequality, Young's inequality and Poincare inequality, we deal with the terms in (2.8) one by one as follow, ( ) By using Holder inequality, Young's inequality, and (H 2 ), we obtain Then, we have , by using Gronwall inequality,we obtain ( ) ( ) ( ) From (H 1 ): ( ) , according to Embedding Theorem, then , then we have So, there exists 0 E and ( ) In addition to the assumptions of Lemma 1, if (H 3 ): ( ) , and ( ) , .
Using Holder inequality, Young's inequality and Poincare inequality, we deal with the terms in (2.29) one by one as follow, ( ) By using Holder inequality, Young's inequality, and (H 1 ), (H 3 ), we obtain . 4 By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get : .
By using the same inequality, we can obtain By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get : .
Then, by using Young's inequality, we have where 4  2 .
, so  is the first eigenvalue of the −∆ , and when 2 N ≥ , ( ) ( ) Then the problem (1.1)-(1.3)exists a unique smooth solution , 0 0, , 0 0 t w x w x w x w x = = = = and the two equations subtract and obtain .
By the Galerkin method and Lemma 1, we can easily obtain the existence of Solutions.Next, we prove the uniqueness of Solutions in detail.Assume , u v are two solutions of (1.1)-(1.3),let w u v = − , then

Global Attractor 3.1. The Existence and Uniqueness of Solution
) We get the uniqueness of the solution.So the proof of the Theorem 3.1.has been completed.t is a completely continuous operator.Therefore, the semigroup operators S(t) exist a compact global attractor A. Theorem 3.3 Under the assume of Theorem 3.1, equations have global attractor [10].Global AttractorTheorem 3.2.[10]LetX be a Banach space, and S