Abstract and Applied Analysis
Volume 2005 (2005), Issue 8, Pages 901-919
doi:10.1155/AAA.2005.901

Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary

M. L. Santos,1 J. Ferreira,2 and C. A. Raposo2

1Departamento de Matemática, Universidade Federal do Pará, Campus Universitário do Guamá, Rua Augusto Corrêa 01, Pará CEP 66075-110, Brazil
2Departamento de Matemática, Estatística e Ciências da Computaçäo, Universidade Federal de São João del-Rei (UFSJ), Praça Frei Orlando 170, São João del-Rei CEP 36300-000, Minas Gerais, Brazil

Received 2 October 2003

Copyright © 2005 M. L. Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the exponential decay in the case n>2, as time goes to infinity, of regular solutions for the nonlinear beam equation with memory and weak damping utt+Δ2u−M(‖∇u‖L2(Ωt)2)Δu+∫0tg(t−s)Δu(s)ds+αut=0 in Q^ in a noncylindrical domain of ℝn+1(n≥1) under suitable hypothesis on the scalar functions M and g, and where α is a positive constant. We establish existence and uniqueness of regular solutions for any n≥1.