Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T10:29:49.602Z Has data issue: false hasContentIssue false

The asymptotics of extinction in nonlinear diffusion reaction equations

Published online by Cambridge University Press:  17 February 2009

R. E. Grundy
Affiliation:
Department of Mathematical Sciences, University of St. Andrews, Scotland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we consider the asymptotics of extinction for the nonlinear diffusion reaction equation

with non-negative initial data possessing finite support. For t > 0, both solution and support vanish as t → T and x → x0. With T as the extinction time we construct the asymptotic solution as τ = T – t → 0 near the extinction point x0 using matched expansions. Taking x0= 0, we first form an outer expansion valid when η =xt–(m–p)/2 (1–p) = 0(1). This is nonuniformly valid for large |η| and has to be replaced by an intermediate expansion valid for |x| = O(τ−1/l0) where l0 is an even integer greater than unity. If p + m ≥ 2 this expansion is uniformly valid while for p + m < 2, there are regions near the edge of the support where diffusion becomes important. The zero order solution in these inner regions is discussed numerically.

Information

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1992