Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation

Yaojun Ye

doi:10.1186/1029-242x-2013-195

Abstract

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equation with dissipative term in a bounded domain, where and are constants. We obtain the global existence of solutions by constructing a stable set in and show the energy decay estimate by applying a lemma of Komornik. MSC:35B40, 35L70.

Highlights

1 Introduction In this paper, we investigate the existence and asymptotic stability of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with nonlinear dissipative term in a bounded domain utt – φ
We prove the global existence for the problem ( . )-( . ) by applying the potential well theory introduced by Sattinger [ ] and Payne and Sattinger [ ]
In order to prove the existence of global solutions of the problem ( . )-( . ), we need the following lemma

Summary

Introduction

We investigate the existence and asymptotic stability of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with nonlinear dissipative term in a bounded domain utt – φ. The nondegenerate case with α = , a > and b = was considered by De Brito, Yamada and Nishihara [ – ], they proved that for small initial data (u , u ) ∈ (H ( ) ∩ H ( )) × H ( ) there exists a unique global solution of When φ(s) ≥ , Ghisi and Gobbino [ ] proved the existence and uniqueness of a global solution u(t) of the problem They proved that, if the initial datas are small enough, the problem

Dα u

If u