Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption
type: Item
instance of: scholarly article
title: Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption
zbMATH Open document ID: 1019.35052
DOI: 10.1016/S0362-546X(01)00764-7
author: Ugur G. Abdulla
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The paper deals with the Cauchy problem for the reaction-diffusion equation \[ u_t - (u^m)_{xx} + bu^{\beta} = 0, \quad u(x,0) = u_0(x) \qquad (x \in \mathbb{R}, 0<t<T). \] The short-time behaviour of the interface function \(\eta(t) = \sup\{x: u(x,t) > 0 \}\) and of the local solution near the interface are studied in the case \(0<m<1\) (fast diffusion). This paper continues earlier investigations of the author together with \textit{J. R. King} [SIAM J. Math. Anal. 32, 235-260 (2000; Zbl 0964.35086)], where the case \(m>1\) (slow diffusion) was treated. The methods used are based on the results concerning the general theory of initial-boundary value problems for reaction-diffusion equations, obtained previously by the author [J. Differ. Equations 164, 321-354 (2000; Zbl 0956.35069)].
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1788029
zbMATH Keywords: reaction-diffusion equations
zbMATH Keywords: nonlinear degenerate parabolic equations
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
cites work: Reaction-diffusion in irregular domains
cites work: Interface Development and Local Solutions to Reaction-Diffusion Equations
cites work: On the initial growth of interfaces in reaction–diffusion equations with strong absorption
cites work: On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium
cites work: On a class of similarity solutions of the porous media equation
cites work: Short time behaviour of a singular solution to the heat equation with absorption
cites work: THE INITIAL INTERFACE DEVELOPMENT FOR A REACTION-DIFFUSION EQUATION WITH POWER-LAW INITIAL DATA
cites work: The Cauchy Problem for u t = Δu m When 0 < m < 1
cites work: Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations
cites work: On solutions of some doubly nonlinear degenerate parabolic equations with absorption
cites work: Behaviour of the Velocity of One-Dimensional Flows in Porous Media
full work available at URL: s0362-546x(01)00764-7
OpenAlex ID: W1976829773
description: scientific article; zbMATH DE number 1788029