Long-time behavior of a quasilinear viscoelastic equation with past history

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Araujo, Rawlilson de Oliveira¹  Ma, To Fu¹  Qin, Yuming² 

[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13566590 Sao Carlos, SP, Brazil

[2] Donghua Univ, Dept Appl Math, Shanghai 201620, Peoples R China

关键词: Quasilinear wave equation;    Viscoelasticity;    Memory;    Exponential stability;    Global attractors;    Uniqueness;   

DOI  :  10.1016/j.jde.2013.02.010

来源: Elsevier

【 摘 要 】

This paper is concerned with a class of quasilinear wave equations with memory vertical bar u(t)vertical bar(rho)u(tt) - alpha Delta u - Delta u(tt) + integral(t)(tau) mu(t - s)Delta u(s)ds - gamma Delta u(t) + f(u) = h, rho > 0, which was considered by several authors, with tau = 0, since 2001. Existing results are mainly devoted to global existence, energy decay, existence with small data and blow-up of,solutions. However uniqueness seems to be an open problem and existence of attractors was no yet considered. The objective of the present paper is to provide some results on the well-posedness and longtime behavior to this equation in a more general setting which includes past history, that is, by taking tau = -infinity in the memory term. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free