ABSTRACT

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: Fir- 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.

Keywords:

Kirchhoff Wave Equation, Global Attractor, The Smoothing Effect, The Regularity, Approximate Inertial Manifold

1. 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 naturally 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:

(1.1)

(1.2)

(1.3)

where is a bounded domain in with smooth boundary, and are positive constants, and the assumptions on 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 and the dimension estimation of glo- bal attractor, have been researched.

In [13] , Dai Zhengde, Guo Boling, Lin Guoguang studied the fractal structure of attractor for the generalized Kuramoto-Sivashinsky equations:

(1.4)

(1.5)

(1.6)

where.

In [14] , Li Yongsheng, Zhang Weiguo studied regularity and approximate of the attractor for the strongly damped wave equation:

(1.7)

(1.8)

(1.9)

where are positive constants.

Luo Hong, Pu Zhilin and Chen Guanggan [15] studied regularity of the attractor and approximate inertial manifold for strongly damped nonlinear wave equation:

(1.10)

(1.11)

(1.12)

where is a positive constant.

Wang Lei, Dang Jinbao and Lin Guoguang [16] also studied the approximate inertial manifolds of the fractional nonlinear Schrödinger equation:

(1.13)

(1.14)

(1.15)

where is a standard orthogonal base,

i is the imaginary unit..

Recently, Sufang Zhang, Jianwen Zhang [17] studied approximate inertial manifold of strongly damped wave equation:

(1.16)

(1.17)

(1.18)

where is a bounded domain in with smooth boundary, is a constant, 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 [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 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).

2. Statement of Some Assumptions, Notations and Main Results

For convenience, we denote the norm and scalar product in by and;

, , , , ,.

Let, where is a bounded domain, where the norm is defined as. is an unbounded positive definite self adjoint operator. Let , From reference [25] , is compact, is dense in E, so, where E is space by as base generated. , where, are the eigenvalues and eigenvectors of A, ,. Then consists of a set of standard orthogonal basis space E.

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

(G₁) From reference [12] , we set some constants:,

such that,.

(G₂) Let, and,.

Theorem 2.1 From reference [12] , due to (G₁), (G₂) hold,

(i) Let, then for each, , the problems (1.1)-(1.3) exist solution u,;,.

(ii) Let, is the semigroup operator for the problems (1.1) - (1.3), then the semigroup exists a compact global attractor. So we can find a compact connected invariant set B to absorb all the bounded sets on.

3. 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.

Let, then the problem (1.1) can be reduced to the following form:

(3.1)

(3.2)

Let

(3.3)

where.

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

(3.4)

From references [26] [27] , is a linear dense closed operator on, which is a sector operator and has a bounded inverse. generates an analytic semigroup on.

Lemma 3.1 From references [14] [15] , due to (G₁), (G₂) hold, let, then

Each, the solution to the problems (1.1) - (1.3) meet the following conditions:

(3.5)

And there exist such that the following inequalities are established:

(3.6)

where, is independent of the initial value.

Proof. By the first conclusion (i) of theorem 2.1, when, the solution u meet:, ,. By the second conclusion (ii) of theorem 2.1, there exist, when,

(3.7)

Meanwhile, is uniformly bounded in E,.

(3.8)

Then,.

Based on the reference [27] , the analytic properties of the semigroups generated by and the Equation (3.4), immediately get, the solution, furthermore, for the non-homogeneous term in the Equation (3.4), , then, , due to are arbitrary, ,.

Since, we are now considering, respectively, as the initial time, initial value. Next, we consider the equation about,

(3.9)

then

(3.10)

(3.11)

(3.12)

(3.13)

Next, we multiply with both sides of the equation (3.10) and integrate over to obtain

(3.14)

(3.15)

(3.16)

where from the hypothesis (G2),

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

where.

By using Gagliardo-Nirenberg’s embedding inequality, Hölder’s inequality:

(3.22)

Similar to the relation (3.20):

(3.23)

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

(3.24)

(3.25)

(3.26)

(3.27)

In reference [12] , are bounded by a priori estimates.

(3.28)

So we get:

(3.29)

From above, we have

(3.30)

(3.31)

Taking, then

(3.32)

(3.33)

At last, we get:

(3.34)

Let;;

;

.

By using Poincaré’s inequality, we get

(3.35)

We take proper, such that:

Then

(3.36)

From the relation (3.36), we can get

(3.37)

By using Gronwall’s inequality, we obtain:

(3.38)

Taking, such that, then

(3.39)

where

(3.40)

Meanwhile, we once again take proper, such that:

So there are, which make the following inequalities:

(3.41)

where, is independent of the initial value.

Lemma 3.2 From references [14] [15] , due to (G₁), (G₂) hold, let, , then, the solution to the problems (1.1) (1.3) meet the following conditions:

(3.42)

And there exist, such that the following inequalities are established:

(3.43)

Proof. Take proper T, such that, , we are now considering the Equation (3.9), assume (G₁), (G₂) hold, , , , the nonlinear term. Based on the reference [27] , the solution to the Equation (3.9):. From (3.4), we get , due to are arbitrary, ,

, and then we can get, ,.

Similar to lemma (3.1), we are now considering, respectively, as the initial time, initial value. Next, and once again, we consider the Equations (3.9) - (3.13), multiplying with both sides of the equation (3.10) and integrating over to obtain

(3.44)

(3.45)

(3.46)

where from the hypothesis (G2),

Similar to lemma 3.1

(3.47)

(3.48)

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

Through similar methods above

From above, we have

(3.49)

(3.50)

Taking, then

At last, we get:

(3.51)

Let;;

;

.

By using Poincaré’s inequality, we get

(3.52)

We take proper, , , , , , , such that:

Then

(3.53)

From the relation (3.53), we can get

(3.54)

By using Gronwall’s inequality, we obtain:

(3.55)

Taking, such that, then

(3.56)

where

(3.57)

Meanwhile, we once again take proper, , such that:

So there are, which make the following inequalities:

(3.58)

where is independent of the initial value.

Similar to above discussions, there are, , which make the following inequalities:

(3.59)

where is independent of the initial value.

Using the original Equation (1.1), we obtain

(3.60)

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

(3.61)

where is independent of the initial value.

So there are, , which make the following inequalities:

(3.62)

where is independent of the initial value.

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

Theorem 3.1 From reference [14] , let is the semigroup operator for the pro- blems (1.1) - (1.3), then the semigroup exists a compact global attractor in, and.

The proof of theorem 3.1 see ref. [14] , is omitted here.

4. The Approximate Inertial Manifold for the Global Attractor

In this section, we first construct a smooth manifold, and then prove that is an approximate inertial manifold of the semigroup, namely, the arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.

Let, is an orthogonal projection from the space E to the subspace spanned by, , so that u is decomposed as the sum.

For the solution u of the problems (1.1) - (1.3), let, , ,. Then, , ,. We use and to act the problem (1.1) respectively.

(4.63)

(4.64)

Let,. Then the problems (4.63) - (4.64) can be written as:

(4.65)

(4.66)

From above, we have, there exist, , is independent of the initial value, and then, So for, , we obtain

(4.67)

Theorem 4.1 From references [14] [15] [16] , according to lemmas 3.1, 3.2 and the theorem 3.1, let is the N dimensional linear subspace of , there exists, is sufficiently large. When, arbitrary trajectory arising from the for the Kirchhoff wave equations, which track into a sphere in. Namely,. Meanwhile, the is called a N dimensional flat approximate inertial manifold of the semigroup.

Remark 4.1. For the problem (4.66), if we do not consider and contained in the nonlinear terms, for, we define mapping:. is the solution of the Equation (4.68):

(4.68)

Then: is a smooth map, its image is , which is a approximate inertial manifold of the semigroup.

Theorem 4.2 From references [14] [15] [16] , according to lemmas 3.1, 3.2 and the theorems 3.1, 4.1, then, there exists, when, arbitrary trajectory arising from the for the Kirchhoff wave equations, which track into a neighborhood in. Namely,. Meanwhile, the is a approximate inertial manifold of the semigroup. Furthermore, , there exists, is sufficiently large,. When, arbitrary trajectory arising from the for the Kirchhoff wave equations, which track into a neighborhood in. Namely,. The is a very precise approximate inertial manifold of the semigroup.

Proof. Firstly, let, then, are the solutions of the problems (4.65) - (4.66), and then let..

From the relation (4.68), we can obtain:

(4.69)

(4.70)

Then from the hypothesis (G₁),.

(4.71)

(4.72)

(4.73)

We put into the relation (4.68), the following relations can be obtained immediately,

(4.74)

(4.75)

Therefore

(4.76)

(4.77)

Then

(4.78)

So, we obtain

(4.79)

A similar method in reference [14] , we immediately get the semigroup exists a compact global attractor in, and, and then , there exists, is sufficiently large,. When, arbitrary trajectory arising from the for the Kirchhoff wave equations, which track into a neighborhood in.

(4.80)

where the is a smooth manifold that we construct, which is very precise, to approximate inertial manifold of the semigroup.

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).

Acknowledgements

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).

Cite this paper

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

References