A parabolic free boundary problem with double pinning
type: Item
instance of: scholarly article
title: A parabolic free boundary problem with double pinning
zbMATH Open document ID: 1072.35210
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: For each \(\epsilon\in (0,1)\) the initial value problem \[ \partial_t u_{\epsilon} -\Delta u_{\epsilon}=-\beta_{\epsilon}(u_{\epsilon})\,\,\, {\text{in}}\,\,\,(0,\infty)\times\mathbb{R}^n \] \[ u_{\epsilon}(0,\cdot)=u^0_{\epsilon}\,\,\,\text{in }\mathbb{R}^n \] is considered with \(\beta_{\epsilon}(z)=\epsilon^{-1}\beta(z/\epsilon)\) and \(\beta\in C'_0([a,b])\) having the following property: it possesses only a finite number of zeroes in \((a,b)\) and not more than one in which it changes from negative to positive. In addition it is assumed that for each \(\epsilon\) the difference \((u^0_{\epsilon}-\epsilon a)\) is bounded in \(C^{(0,1)}(\mathbb{R}^M)\) with uniformly bounded support, and it has a limit \(u^0\leftarrow H^{1,2}(\mathbb{R}^n)\) as \(\epsilon\rightarrow 0\). The author proves that (i) the set of the solutions \(u_{\epsilon}\) is compact in the sense that \(u_{\epsilon}-\epsilon a\) is bounded in \(L^{\infty}[(0,\infty);H^{1,2}_{loc}(\mathbb{R}^n)]\) and \(\partial_t u_{\epsilon}\) is bounded in \(L^2_{loc}([0,\infty)\times\mathbb{R}^n)\) and that \(u_{\epsilon}\) is also bounded in \(C^{1/2}([0,T]\times\mathbb{R}^n)\) for all bounded \(T\), (ii) each limit of \(u_{\epsilon}\) is (in a suitable sense) a solution of the free boundary problem \[ \partial_t u-\Delta u=0\,\,\, \text{in}\,\,\,(\{u>0\}\cup\{u<0\})\cap ((0,\infty)\times\mathbb{R}^n), \] \[ | \nabla u^+| ^2-| \nabla u^-| ^2=2\int^b_a \beta (s)\,ds \] on the interface.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 2102563
zbMATH Keywords: singularities
author: Georg S. Weiss
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.005
OpenAlex ID: W2013015161
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DOI: 10.1016/J.NA.2004.02.005
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description: scientific article; zbMATH DE number 2102563