The Inertial Manifold for Class Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms

Guoguang Lin,Xiangshuang Xia

doi:10.4236/am.2018.96050

Abstract

In this paper, we study the inertial manifolds for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The inertial manifold is a finite dimensional invariant smooth manifold that contains the global attractor, attracting the solution orbits by the exponential rate. Under appropriate assumptions, we firstly exert the Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, and then we prove the existence of the inertial manifold by showing that the spectral gap condition is true.

Highlights

In this paper, we concerned the equation:( ) utt − M ∇u 2 + ∇mv 2 ∆u − β∆ut + g1 (u, v) =f1 ( x), ( ) vtt + M∇u 2 + ∇mv 2 (−∆)m v + β (−∆)m vt + g2 (u, v) = f2 ( x),= u ( x, 0) u0 ( x), = ut ( x, 0) u1 ( x), x ∈ Ω, (1.1)= v ( x, 0) v0 ( x), = vt ( x, 0) v1 ( x), x ∈ Ω, u = ∂Ω 0, v= ∂Ω 0
We study the inertial manifolds for a class of the Kirchhoff-type equations with strongly damped terms and source terms
We firstly exert the Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, and we prove the existence of the inertial manifold by showing that the spectral gap condition is true

Summary

Introduction

The study on the complexity of the space-time of high dimensional and infinite dimensional dynamical systems has gradually become the focus of nonlinear scientific research. The inertial manifold has been found in the researches of the long time behavior of the solution and the attractor structure. The inertial manifold is a powerful tool to study the long-time behavior of nonlinear dissipative systems and expose the real or seemingly chaotic structure of nonlinear dynamics. In 2010, Guoguang Lin and Jingzhu Wu [3] studied the existence of the inertial manifold of Boussinesq equation: uut(t. In 2016, Ling Chen, Wei Wang and Guoguang Lin [4] established the exponential attractors and inertial manifolds of the higher-order Kirchhoff-type equation:.

Preliminaries

The Inertial Manifold