The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

Guoguang Lin,Xiaomei Liu

doi:10.4236/jamp.2022.101013

Abstract

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

Highlights

The Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and the existence of its exponential attractor is obtained
Exponential attractor is a compact positive invariant set with finite fractal dimension and exponentially attracts every orbit, which is an important feature to describe the long-term behavior of nonlinear partial differential equations
Inertial manifold refers to the positive invariant Lipschitz manifold of finite dimension, which includes the global attractor attracting all solution orbits at exponential speed, and it is an important bridge between infinite dimensional dynamical system and finite dimensional dynamical system

Summary

Introduction

Exponential attractor is a compact positive invariant set with finite fractal dimension and exponentially attracts every orbit, which is an important feature to describe the long-term behavior of nonlinear partial differential equations. In reference [2], the author studied the exponential attractors of the following nonlinear wave equations by using operator decomposition and finite covering methods. Studied the global existence and blow-up of solutions for the following high-order Kirchhoff type equations with nonlinear dissipation terms. Salim [5] improves the results in reference [4] by modifying the proof method, and proves that when the positive initial energy has an upper bound, the solution explodes in a finite time. Inspired by the above research, this paper will discuss a family of the existence of exponential attractors and inertial manifolds of a generalized Kirchhoff equation with damping term:.

Preliminaries

Exponential Attractors

A Family of Inertial Manifolds