Global existence and blowup for a parabolic equation with a non-local source and absorption
type: Item
instance of: scholarly article
title: Global existence and blowup for a parabolic equation with a non-local source and absorption
zbMATH Open document ID: 1270.35248
author: Yanyan Li
published in: Acta Applicandae Mathematicae
publication date: 2013-04-11T00:00:00Z
review text: The authors consider the following one-dimensional free boundary problem for a parabolic equation with a non-local source and absorption: \[ u_{t}-du_{xx}=\int_{g(t)}^{h(t)}u^{p}(t,x)dx-ku^{q}(t,x),\quad t>0,\;g(t)<x<h(t), \] \[ u(t,g(t))=0,\quad g'(t)=-\mu u_{x}(t,g(t)),\quad t>0 \] \[ u(t,h(t))=0,\quad h'(t)=-\mu u_{x}(t,h(t)),\quad t>0 \] \[ g(0)=-h_{0},\quad h(0)=h_{0},\quad u(0,x)=u_{0}(x),\quad -h_{0}\leq x\leq h_{0}, \] where both \(x=g(t)\) and \(x=h(t)\) are moving boundaries to be determined together with the solution \(u(t,x).\) Here \(h_{0}>0,\) \(p>q>1,\) \(d,k\) and \(\mu\) are positive constants, and the initial function \(u_{0}\) is a non-negative non-trivial function satisfying \(u_{0}\in C^{2}([-h_{0}h_{0}]),\) \(u_{0}(-h_{0})=u_{0}(h_{0})=0\) and \(u_{0}>0\) in \((-h_{0},h_{0}.)\) The local existence and uniqueness is investigated. The description of the free boundaries is given. As main result of the blowup property discussion consists in the proof of the existence of a global fast solution and a global slow solution.
reviewed by: Svetlana A. Grishina
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 6153126
zbMATH Keywords: one space dimension
zbMATH Keywords: global fast solution
zbMATH Keywords: global slow solution
Wikidata QID: Q61635620
MaRDI profile type: Publication
full work available at URL: s10440-012-9774-3
OpenAlex ID: W2087189305
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DOI: 10.1007/S10440-012-9774-3
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description: scientific article; zbMATH DE number 6153126