Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms

This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: ( ) ( ) 1 2 1 1 | | . p q tt t t t u u u u u u u u f x ε α β φ − − − ∆ + + − ∇ ∆ = Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.

stly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space.Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation.Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.

Introduction
It is well known that we are studying the long time behavior of the infinite dimensional dynamical systems of the nonlinear partial differential equations, and the concept of the inertial manifold plays an important role in this field.In 1985, G. Foias, G. R. Sell and R. Teman [1] first put forward the concept of the inertial manifold; it is an invariant finite dimensional Lipschitz manifold; it is exponentially attracting trajectory and contains the global attractor.But to ensure that existing conditions are very harsh for inertial manifolds (For instance, spectral interval condition), the existence of a large number of important partial differential equations is still not solved.Therefore, people na-C.F. Ai et al. turally think of using an approximate, smooth and easy to solve the manifolds to approximate the global attractor and inertial manifolds, which is the approximate inertial manifold.
Approximate inertial manifolds are finite dimensional smooth manifolds, and each solution of the equation is in a finite time to its narrow field.In particular, the global attractor is also included in its neighbourhood.The existence of approximate inertial manifolds of a large number of dissipative partial differential equations has been studied [2]- [7].
In this paper, we are concerned a class of the Kirchhoff wave equations with nonlinear strongly damped terms referred to as follows: ( ) ( ) ,0 ; ,0 , , where Ω is a bounded domain in N  with smooth boundary ∂Ω , and 1 , , ε α β are positive constants, and the assumptions on ( ) 2 u φ ∇ will be specified later.
In [8], G. Kirchhoff firstly proposed the so called Kirchhoff string model in the study nonlinear vibration of an elastic string.Kirchhoff type wave equations have been studied by many scholars (see [9] [10] [11]).In reference [12], the long time behavior of solutions for the initial value problems (1.1) -(1.3), the existence of global attractor corresponding to the semigroup operator ( ) S t and the dimension estimation of global attractor, have been researched.
Luo Hong, Pu Zhilin and Chen Guanggan [15] studied regularity of the attractor and approximate inertial manifold for strongly damped nonlinear wave equation: where α is a positive constant.
Wang Lei, Dang Jinbao and Lin Guoguang [16] also studied the approximate inertial manifolds of the fractional nonlinear Schrödinger equation: where Recently, Sufang Zhang, Jianwen Zhang [17] studied approximate inertial manifold of strongly damped wave equation: where Ω is a bounded domain in N  with smooth boundary ∂Ω , 0 α > is a con- stant, the function ( ) There have many researches on approximate inertial manifolds for nonlinear wave equations (see [18]- [24]).In order to construct the approximate inertial manifolds for the initial boundary value problems, in the references [14] to [15], the regularity of the global attractor is studied, and then the approximate inertial manifold is constructed.In The paper is arranged as follows.In Section 2, we state some assumptions, notations and the main results are stated.In Section 3, through the estimation of solution smoothness of higher order, then we obtain the regularity of the global attractor.In Section 4, by constructing a smooth manifold, namely the approximate inertial manifold, we approximate the global attractor for the problems (1.1) -(1.3).

Statement of Some Assumptions, Notations and Main Results
For convenience, we denote the norm and scalar product in L Ω by .and ( ) .,. ; ( ) N Ω ⊂  is a bounded domain, where the norm is defined as . .A = −∆ is an unbounded positive definite self adjoint operator.Let ( ) ( ) ( ) , where E is space by { } λ , k ω are the eigenvalues and eigenvectors of A, ⋅⋅⋅ .Then k ω consists of a set of standard or- thogonal basis space E.
We present some assumptions and notations needed in the proof of our results as follows: (G 1 ) From reference [12], we set some constants: Theorem 2.1 From reference [12], due to (G 1 ), (G 2 ) hold,

The Regularity of Global Attractor
In order to obtain the regularity of global attractor, we need to give a higher order uniform a priori estimates for the solution.
Based on the reference [27], the analytic properties of the semigroups generated by Λ and the Equation (3.4), immediately get tively, as the initial time, initial value.Next, we consider the equation about Next, we multiply t v v ε + with both sides of the equation (3.10) and integrate over Ω to obtain ( ) ( ) , , , where from the hypothesis (G2), where . By using Gagliardo-Nirenberg's embedding inequality, Hölder's inequality: Similar to the relation (3.20): ( ) ( ) By using Hölder's inequality, Young's inequality and Sobolev's embedding inequality: ; 2 In reference [12], , u u ∆ are bounded by a priori estimates.
Meanwhile, we once again take proper ε , 1 ε , such that: > , which make the following inequalities: ( ) ( ) where 1 R is independent of the initial value 0 U .
Similar to above discussions, there are R > , which make the following inequalities: ( ) ( ) , , .
where 2 R is independent of the initial value 0 U .
Using the original Equation (1.1), we obtain Next, using the elliptic property of the operator A, we get: where 3 R is independent of the initial value 0 U .
According to Lemmas 3.1, 3.2, we can get the following theorem : Theorem 3.1 From reference [14], let ( ) The proof of theorem 3.1 see ref. [14], is omitted here.

The Approximate Inertial Manifold for the Global Attractor
In this section, we first construct a smooth manifold ( ) , and then prove that 1  is an approximate inertial manifold of the semigroup ( ) S t , namely, the arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time. Let P is an orthogonal projection from the space E to the subspace spanned by so that u is decomposed as the sum u p q = + .
For the solution u of the problems (1.1) -(1.3), let We use . Then the problems (4.63) -(4.64) can be written as:  ( ), From above, we have )   ( ) , which is a approximate inertial manifold of the semigroup ( ) S t .where the n  is a smooth manifold that we construct, which is very precise, to ap- proximate inertial manifold of the semigroup ( )

[ 18 ]
, Tian Lixin, Lin Yurui construct approximate inertial manifolds under spline wavelet basis in weakly damped forced KdV equation.In infinite-dimensional dynamical systems, Kirchhoff type wave equation is a class of very important equation.However, the approximate inertial manifold and inertial manifold of the Kirchhoff wave equation with nonlinear strong damping term are rarely studied.Based on the current research situation of Kirchhoff wave equations, in this paper, we first study the regularity of the global attractor for a class of the Kirchhoff wave equations with nonlinear strongly damped terms, and then construct its approximate inertial manifold.
the solution to the problems (1.1) -(1.3) meet the follow-ing conditions: the solution to the problems (1.1) (1.3) meet the following conditions: , as the ini- tial time, initial value.Next, and once again, we consider the Equations (3.9) -(3∆ with both sides of the equation (3.10) and integrating over Ω t is the semigroup operator for the problems (1.1) -(1.3), then the semigroup ( ) S t exists a compact global attractor 1

N
P and NQ to act the problem (1.1) respectively.

Theorem 4 . 2 1 
From references[14] [15][16], according to lemmas 3.1, 3.2 and the theorems 3trajectory arising from the 0 U for the Kirchhoff wave equations, which track into a is a approximate inertial manifold of the semigroup ( ) S t .Furthermore, .Ai et al.

Remark 4 . 2 .
This article is based on the references[14] [15][16], by estimating the higher regularity of the global attractor, then we construct its approximate inertial manifold.Approximate inertial manifold, which is a kind of nonlinear, finite dimensional and has certain smoothness.It is of great significance to study the long time behavior of the dissipative equations and the structure of the attractors.On the basis of this article, then we are likely to consider the inertial manifold of the global attractor for the problems (1.1) -(1.3).
flat approximate inertial manifold of the semigroup ( ) is a very precise approximate inertial manifold of the semigroup ( ) t into the relation (4.68), the following relations can be obtained im- W