Asymptotically self-similar global solutions of a damped wave equation with nonlinear memory

Article type: Research Article

Authors: Berbiche, Mohamed

Affiliations: Département de Mathématiques, Université de Khenchela Route de Constantine, Bp: 1252, El-Houria 40004, Khenchela, Algérie. E-mail: berbichemed@yahoo.fr

Abstract: In this paper we consider the Cauchy problem of semi-linear damped wave equation with nonlinear memory term a∫0t(t−τ)−γ|u|α−1u(τ,x) dτ. We prove global existence and asymptotic behavior of solution for small initial data. Moreover, we show that the global solutions behave asymptotically like self-similar solutions of the semi-linear heat equation with nonlinear memory as t→∞.

Keywords: damped wave equation, self-similar solutions, semi-linear heat equation

DOI: 10.3233/ASY-2012-1147

Journal: Asymptotic Analysis, vol. 82, no. 3-4, pp. 315-330, 2013