A non-degeneracy property for a class of degenerate parabolic equations
type: Item
instance of: scholarly article
title: A non-degeneracy property for a class of degenerate parabolic equations
zbMATH Open document ID: 0858.35069
DOI: 10.4171/ZAA/720
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-10-14T00:00:00Z
review text: Summary: We deal with the initial and boundary value problem for the degenerate parabolic equation \(u_t=\Delta\beta(u)\) in the cylinder \(\Omega\times (0,T)\), where \(\Omega\subset \mathbb{R}^n\) is bounded, \(\beta(0)= \beta'(0)=0\), and \(\beta'\geq 0\) (e.g., \(\beta(u)= u|u|^{m-1}\) \((m>1)\)). We study the appearance of the free boundary, and prove under certain hypothesis on \(\beta\) that the free boundary has a finite speed of propagation, and is Hölder continuous. Further, we estimate the Lebesgue measure of the set where \(u>0\) is small and obtain the non-degeneracy property \(|\{0< \beta'(u(x,t))< \varepsilon\}|\leq c\varepsilon^{1/2}\).
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 935176
zbMATH Keywords: speed of propagation
zbMATH Keywords: porous medium equations
author: C. Ebmeyer
MaRDI profile type: Publication
cites work: Q4180668
cites work: Interfaces with a corner point in one-dimensional porous medium flow
cites work: Continuity of the Density of a Gas Flow in a Porous Medium
cites work: Q3882115
cites work: Q3786839
cites work: Q4377831
cites work: Hölder continuity of interfaces for the porous medium equation with absorption
cites work: The Porous Medium Equation in One Dimension
full work available at URL: https://doi.org/10.4171/zaa/720
OpenAlex ID: W2046639127
description: scientific article; zbMATH DE number 935176