Journal of Applied Mathematics 
Volume 2003 (2003), Issue 8, Pages 409-427
doi:10.1155/S1110757X03212067

Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

Robert Willie1,2

 1Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain
2Department of Mathematics, University of Zimbabwe, Harare M.P. 167, Zimbabwe
Received 20 December 2002

Abstract

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.