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

Chengfei Ai,Guoguang Lin,Huixian Zhu

doi:10.4236/ijmnta.2016.54020

Chengfei Ai, Guoguang Lin + Show 1 more

Open Access

https://doi.org/10.4236/ijmnta.2016.54020

Copy DOI

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: . 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.

Highlights

We obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation
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

Summary

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. We are concerned a class of the Kirchhoff wave equations with nonlinear strongly damped terms referred to as follows:. 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 the approximate inertial manifold is constructed. 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 construct its approximate inertial manifold. We can find a compact connected invariant set B to absorb all the bounded sets on D ( A)× E

The Regularity of Global Attractor

The Approximate Inertial Manifold for the Global Attractor