The Exponential Attractor for a Class of Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms

Guoguang Lin,Xiangshuang Xia

doi:10.4236/jamp.2018.67125

Abstract

This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The exponential attractor is also called the inertial fractal set, which is an intermediate step between global attractors and inertial manifolds. Obtaining a set that attracts all the trajectories of the dynamical system at an exponential rate by the methods of Eden A. Under appropriate assumptions, we firstly construct an invariantly compact set. Secondly, showing the solution semigroups of the Kirchhoff-type equations is squeezing and Lipschitz continuous. Finally, the finite fractal dimension of the exponential attractor is obtained.

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)= v ( x, 0) v0 ( x), = vt ( x, 0) v1 ( x), x ∈ Ω, u = ∂Ω 0, v= ∂Ω 0
This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms
The exponential attractor is called the inertial fractal set, which is an intermediate step between global attractors and inertial manifolds

Summary

Introduction

The exponential attractor is a positively invariant compact set with finite fractal dimensions and attracts the solution orbit at an exponential rate It is a tangible concept between the global attractor and the inertial manifold. Eden et al [8] proposed the concept of inertial sets which were compact sets of finite fractal dimension and attracted all the solutions with an exponential rate of convergence as early as in 1990 They showed the long time dynamics of the dissipative evolution equations are characterized by an inertial set. They obtained the global attractor and exponential attractor with finite fractal dimension under appropriate conditions.

Preliminaries

The Exponential Attractor

Conclusion