Viscosity solution theory of a class of nonlinear degenerate parabolic equations. I: Uniqueness and existence of viscosity solutions
type: Item
instance of: scholarly article
title: Viscosity solution theory of a class of nonlinear degenerate parabolic equations. I: Uniqueness and existence of viscosity solutions
zbMATH Open document ID: 0882.35067
DOI: 10.1007/BF02015135
author: Yi Zhan
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1998-02-01T00:00:00Z
review text: For the nonlinear degenerate parabolic equations \[ u_t+F(Du,D^2u)= 0,\;(x,t)\in \mathbb{R}^n\times (0,T), \] with \(F\) satisfying the following assumptions: \(F(q,X)\in C(\mathbb{R}^n\times S^n)\), \(F(q,X+Y)\leq F(q,X)\), for all \(Y\geq 0\), \(F(q,X)\geq -A|q|^{p-2}|X|_1- B|q|^{p-1}\) for all \(A,B\geq 0\) and \(p>2\), where \(S^n\) denotes the space of \(n\times n\) symmetric matrices, the author establishes the comparison principle, regularity, existence, and some geometric properties of the interface for the viscosity solutions. The evolution \(p\)-Laplacian equation and its anisotropic version are typical examples for this class of equations. The approximation method and Perron's method developed by Ishii are used to construct the viscosity solution to the Cauchy problem, and the \(W^{1,1,\infty}\) estimates for initial functions in \(W^{2,\infty}\) are obtained by constructing the global obstacle functions.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1063725
zbMATH Keywords: comparison principle
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: geometric properties of the interface
zbMATH Keywords: global obstacle functions
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1063725