Quantitative unique continuation for the semilinear heat equation in a convex domain
type: Item
instance of: scholarly article
title: Quantitative unique continuation for the semilinear heat equation in a convex domain
zbMATH Open document ID: 1215.35042
author: K.-D. Phung
author: Gengsheng Wang
published in: Journal of Functional Analysis
publication date: 2010-07-19T00:00:00Z
review text: Let \(\Omega\) be a bounded convex smooth domain in \(\mathbb R^d\). The authors study unique continuation properties of solutions of the semilinear heat equation \(u_t-\Delta u=g(u)\) in \(\Omega\times(0,T_*)\) complemented by the homogeneous Dirichlet boundary conditions. Assume first that \(g\) is \(C^1\) and \(g'\) satisfies suitable growth conditions guaranteeing the well-posedness of the problem in \(L^2(\Omega)\). Let \(u,v\) be two solutions with initial data in \(L^2(\Omega)\), let \(T>0\) be less than the minimum of their maximal existence times and \(\omega\subset\Omega\) be open and non-empty. One of the main results of the paper guarantees the estimate \[ \int_\Omega|(u-v)(x,T)|^2\,dx \leq C\biggl(\int_\omega|(u-v)(x,T)|^2\,dx\biggr)^\alpha, \] where \(\alpha\in(0,1)\) depends on \(\Omega\) and \(\omega\) and \(C>0\) depends on \(\Omega,\omega,T\), the norms of the initial data in \(L^2(\Omega)\) and the growth of \(g'\). In addition, if \(u(\cdot,T)=v(\cdot,T)\) over \(\omega\) then \(u=v\) over \(\Omega\times[0,T]\). If \(g\) is merely locally Lipschitz and the initial data are in \(L^\infty(\Omega)\) then the authors prove an estimate of the form \[ \int_\Omega|(u-v)(x,0)|^2\,dx \leq C\int_\omega|(u-v)(x,T)|^2\,dx. \]
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 5757626
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: unique continuation
zbMATH Keywords: homogeneous Dirichlet boundary conditions
zbMATH Keywords: maximal existence times
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.04.015
OpenAlex ID: W2001188635
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DOI: 10.1016/J.JFA.2010.04.015
description: scientific article; zbMATH DE number 5757626