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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: . 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.

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 [

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 [

In this paper, we are concerned a class of the Kirchhoff wave equations with nonlinear strongly damped terms referred to as follows:

where

In [

In [

where

In [

where

Luo Hong, Pu Zhilin and Chen Guanggan [

where

Wang Lei, Dang Jinbao and Lin Guoguang [

where

i is the imaginary unit.

Recently, Sufang Zhang, Jianwen Zhang [

where

There have many researches on approximate inertial manifolds for nonlinear wave equations (see [

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).

For convenience, we denote the norm and scalar product in

Let

We present some assumptions and notations needed in the proof of our results as follows:

(G_{1}) From reference [

such that

(G_{2}) Let

Theorem 2.1 From reference [_{1}), (G_{2}) hold,

(i) Let

(ii) Let

In order to obtain the regularity of global attractor, we need to give a higher order uniform a priori estimates for the solution.

Let

Let

where

Further, we rewrite the problems (1.1) - (1.3):

From references [

Lemma 3.1 From references [_{1}), (G_{2}) hold, let

Each

And there exist

where

Proof. By the first conclusion (i) of theorem 2.1, when

Meanwhile,

Then

Based on the reference [

Since

then

Next, we multiply

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:

In reference [

So we get:

From above, we have

Taking

At last, we get:

Let

By using Poincaré’s inequality, we get

We take proper

Then

From the relation (3.36), we can get

By using Gronwall’s inequality, we obtain:

Taking

where

Meanwhile, we once again take proper

So there are

where

Lemma 3.2 From references [_{1}), (G_{2}) hold, let

And there exist

Proof. Take proper T, such that_{1}), (G_{2}) hold,

Similar to lemma (3.1), we are now considering

where from the hypothesis (G2),

Similar to lemma 3.1

By using Hölder’s inequality, Young’s inequality and Sobolev’s embedding inequality:

Through similar methods above

From above, we have

Taking

At last, we get:

Let

By using Poincaré’s inequality, we get

We take proper

Then

From the relation (3.53), we can get

By using Gronwall’s inequality, we obtain:

Taking

where

Meanwhile, we once again take proper

So there are

where

Similar to above discussions, there are

where

Using the original Equation (1.1), we obtain

Next, using the elliptic property of the operator A, we get:

where

So there are

where

According to Lemmas 3.1, 3.2, we can get the following theorem :

Theorem 3.1 From reference [

The proof of theorem 3.1 see ref. [

In this section, we first construct a smooth manifold

Let

For the solution u of the problems (1.1) - (1.3), let

Let

From above, we have

Theorem 4.1 From references [

Remark 4.1. For the problem (4.66), if we do not consider

Then

Theorem 4.2 From references [

Proof. Firstly, let

From the relation (4.68), we can obtain:

Then from the hypothesis (G_{1}),

We put

Therefore

Then

So, we obtain

A similar method in reference [

where the

Remark 4.2. This article is based on the references [

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper. This work is supported by the Nature Science Foundation of China (No. 11561076).

Ai, C.F., Zhu, H.X. and Lin, G.G. (2016) Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms. International Journal of Modern Non- linear Theory and Application, 5, 218-234. http://dx.doi.org/10.4236/ijmnta.2016.54020