Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations
type: Item
instance of: scholarly article
title: Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations
zbMATH Open document ID: 0876.35063
author: Tamir Tassa
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-10-22T00:00:00Z
review text: The paper deals with uniqueness of piecewise smooth weak solutions to the Cauchy problem for a degenerate parabolic equation of the form \[ u_t -\nabla f=\nabla (Q\nabla u) +g,\quad (x,t)\in \mathbb{R}^N\times \mathbb{R}^+, \] where the flux \(f,\) the viscosity coefficient \(Q,\) and the source term \(g\) depend on \((x,t,u),\) and \(Q=Q(x,t,u)=(Q_{i,j}(x,t,u))^N_{i,j=1}\) is nonnegative definite. In this case, classical solutions as a rule do not exist and, therefore, weak solutions are considered. It is well known that the Cauchy problem has weak solutions. However, due to the possible degeneracy, weak solutions, in general, are not uniquely determined by the initial data. In order to have uniqueness, the author introduces a class of piecewise smooth weak solutions which are \(C^2-\) smooth everywhere apart from a closed nowhere dense collection of smooth manifolds. He proves that the solution operator is \(L^1-\) stable in this class, and as a consequence, the Cauchy problem admits at most one \(L^1(\mathbb{R}^N)\) piecewise smooth weak solution.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1046948
zbMATH Keywords: Degenerate nonlinear parabolic equations
zbMATH Keywords: \(L^ 1-\) stable
zbMATH Keywords: uniqueness
zbMATH Keywords: piecewise smooth weak solutions
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5417
OpenAlex ID: W2087827952
DOI: 10.1006/JMAA.1997.5417
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description: scientific article; zbMATH DE number 1046948