Global attractor for a class of nonlinear generalized Kirchhoff models

The paper studies the long time behavior of solutions to the initial boundary value problem(IBVP) for a class of Kirchhoff models flow ) ( = ) | | (1 ) ( 1 2 2 x f u u u u u u u p t t tt             .We establish the well-posedness, the existence of the global attractor in natural energy space 1 0 1 0 2 ) ( H H H   .

.We establish the well-posedness, the existence of the global attractor in natural energy space

Introduction
In this paper,we are concerned with the existence of global attractor for the following nonlinear plate equation referred to as Kirchhoff models: .
Where  is a bounded domain in N R , 1  p , and  , are positive constants,and the assumptions on ) ( 2 u   will be specified later. Global attractor is a basic concept in the study of the asymptotic behavior of solutions for nonlinear evolution equations with various dissipation. From the physical point of view, the global attractor of the dissipative equation(1.1)represents the permanent regime that can be observed when the excitation starts from any point in natural energy space, and its dimension represents the number of degree of freedom of the related turbulent phenomenon and thus the level of complexity concerning the flow. All the information concerning the attractor and its dimension frim the qualitative nature to the quantitative nature then yield valuable information concerning the flows that this physical system can generate. On the physical and numerical simulations [1].
Many authors have focused on the Kirchhoff equations, Igor Chueshov [2]studied the long-time dynamics of Kirchhoff wave models with strong nonlinear damping: ).
is a bounded domain with smooth boundary   . , w w w . c i r w o r l d . c o m Yang Zhijian[5] also studied the longtime behavior of the Kirchhoff type equation with strong damping on N R : They established the well-posedness, the existence of the global and exponential attractors in natural energy space Claudianor O.Alves and Giovany M.Figueiredo [7]proved the existence of positive solutions for the following class of nonlocal problem: . For more related results, we refer the reader to [8]- [11]. The paper is arranged as follows. In Sec.2, some notations and the main results are stated. In Sec.3, the global existence of solutions to problem In Sec.4, the existence of global attractor for the dynamical system associated with problem

Statement of main results
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.
We define the operator Now, we state the main results of the paper.

Global existence of solutions
We first prepare the following well known lemmas which will be needed later. [2] [11] . If either

Lemma 3.1(Sobolev-Poincare)
Lemma 3.2(Gronwall's inequality) [11] . Let ) (t H be a non-negative absolutely continuous function on By using Holder's inequality, Young's inequality and Poincare's inequality, we deal with the terms in (3.2) one by one as follow , by using Gronwall's inequality, we obtain ), e (1 e (0) ) ( Substituting (3.19), (3.20) into (3.14), we receive where  is a small positive constants. Now we have . := 2 The uniqueness is standard; let Here, we note that the first and second terms in the right-hand side of (3.34) are bounded by One can also find that the last term in the right-hand side of (3.34) is bounded by  [8] [11] The continuous semigroup ) (t S defined on a Banach space X has a global attractor which is connected when the following conditions are satisfied 1) There exists a bounded absorbing set X B  such that for any bounded set    (4.4) where and in the sequel

Bounded absorbing sets and Global attractor in
Let us write now is the solution of (4.6), then 0, ), ( Taking  H inner product by q in (4.10), we have It is clear that is the solution of (4.7), then it exists compact set Proof. Applying Obviously,