The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms

Wei Wang,Guoguang Lin,Ling Chen

doi:10.24297/jam.v12i10.108

Abstract

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -orderkirchhoff-type equation with nonlinear strongly dissipation:At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.

Highlights

There have been many researches on the well-positive and the longtime dynamics for Kirchhoff equation.we can see[1,2,3,4,5,6],FUCAI Li [5] deals with the higher-order kirchhoff-type equation with nonlinear dissipation: u tt
Where R 2 is bounded set; is the bound of ; (u ) and g (u ) are considered as smooth function of u ( x, y, t ) .under the existence of the global solution, it proves that the global attractor and Hausdorff dimensions and fractal dimension
The paper is arranged as follows.in section 2,we state some preliminaries under the assume of Lemma 1and Lemma 2, we get the existence and uniqueness of solution; in section 3,we obtain the global attractors for the problems (1.1)-(1.3);in section 4.we consider that the global attractor of the the above mentioned problems (1.1)-(1.3) has finite Hausdorff dimensions and fractal dimensions

Summary

Introduction

In this paper we concerned with the long time behavior of solution to the initial boundary value problems for Higher-order Kirchhoff-type equation with nonlinear strongly dissipation :. Where R 2 is bounded set; is the bound of ; (u ) and g (u ) are considered as smooth function of u ( x , y , t ) .under the existence of the global solution, it proves that the global attractor and Hausdorff dimensions and fractal dimension. The paper is arranged as follows.in section 2,we state some preliminaries under the assume of Lemma 1and Lemma 2, we get the existence and uniqueness of solution; in section 3,we obtain the global attractors for the problems (1.1)-(1.3);in section 4.we consider that the global attractor of the the above mentioned problems (1.1)-(1.3) has finite Hausdorff dimensions and fractal dimensions

Preliminaries

Global attractor

B H H is a bounded set that includes in the ball