Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations
type: Item
instance of: scholarly article
title: Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations
zbMATH Open document ID: 0645.35100
DOI: 10.1007/BF01774286
author: Martino Bardi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1987-01-01T00:00:00Z
review text: Betrachtet wird das nichtlineare Problem \[ (*)\quad u_ t=\Delta \eta (u)+f(u)\quad in\quad {\mathbb{R}}^ N\times]0,\infty [;\quad u(x,0)=u_ 0(x), \] wobei \(u_ 0\geq 0\) kompakten Träger hat und die nichtlineare Funktion \(\eta\) gewisse Bedingungen erfüllt. Die Menge \(\partial P(t)\), wobei \(P(t):=\{x\in {\mathbb{R}}^ N|\) \(u(x,t)>0\}\), ist der zu (*) gehörige freie Rand. Es wird gezeigt, daß es ein \(t_ 0\) gibt, so daß P(t) für \(t\geq t_ 0\) sternförmig ist. Weiterhin wird eine asymptotische sphärische Symmetrie des freien Randes gezeigt. Dazu sei \(F_ t: \{| \omega | =1\}\to \partial P(t)\) die Abbildung eines Punktes \(\omega\) auf den eindeutig bestimmten gemeinsamen Punkt von \(\partial P(t)\) und \(\{\) \(s\omega|\) \(s>0\}\). Der Autor beweist, daß für \(t\to \infty\) die Funktion \(F_ t\) nach einer Normierung zur identischen Abbildung der Einheitssphäre konvergiert.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 4052186
zbMATH Keywords: free boundary
zbMATH Keywords: Lipschitz continuity
zbMATH Keywords: asymptotic spherical symmetry
zbMATH Keywords: almost radiality
zbMATH Keywords: porous medium equation
zbMATH Keywords: comparison principle
zbMATH Keywords: method of moving planes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4052186