On the regularity at shrinking points for the porous media equation
type: Item
instance of: scholarly article
title: On the regularity at shrinking points for the porous media equation
zbMATH Open document ID: 0948.35070
DOI: 10.1007/PL00001419
author: J. Mossino
author: Maura Ughi
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2000-11-06T00:00:00Z
review text: In the article under consideration, the authors study bounded solutions of the porous media equation in a noncylindrical domain \(Q_T\) in space-time. It is assumed that for any \(\tau\), \(t_0 < \tau < T\), the domain \(Q_{\tau}\) is homeomorphic, with a \(C^2\) mapping, to a cylinder \(\Omega\times (t_0,\tau)\), where \(\Omega\) has the classical regularity in parabolic problems, and also it is supposed that \(Q_T\) ''shrinks'' at \(P\), that is \[ \overline Q_T\cap\{t=T\} = P, \] where \(P\) is the point \((x = 0, t = T)\); \(P\) is called the ''shrinking point'' of \(Q_T\). The main aim of the article is to give conditions on the domain which ensure the continuity or discontinuity of the solutions of the porous media equation at \(P\). The authors use a recent general result which allows comparisons with the heat equation. This result allows to compare for each \(t\) \(\|u(t) - C\|\) with \(\|v(t) - C\|\) where \(C\) is the constant boundary value, \(u\) solves the porous media equation and \(v\) solves a heat equation in a radially symmetric domain whose measure for \(t\) is the same as for a given domain \(\Omega(t)\). The authors obtain a sufficient continuity condition (at \(P\)) for \(u\), in general domains and a sufficient discontinuity condition for \(u\) in radial domains and prove that the continuity of \(u\) is not a local property of \(Q_T\) by means of an explicit counterexample.
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1439280
zbMATH Keywords: time depending domains
zbMATH Keywords: shrinking domains
zbMATH Keywords: noncylindrical domain
zbMATH Keywords: sufficient continuity condition
zbMATH Keywords: sufficient discontinuity condition
MaRDI profile type: Publication
full work available at URL: pl00001419
OpenAlex ID: W2050646739
description: scientific article; zbMATH DE number 1439280