Generalized solutions of parabolic initial value problems with discontinuous nonlinearities
type: Item
instance of: scholarly article
title: Generalized solutions of parabolic initial value problems with discontinuous nonlinearities
zbMATH Open document ID: 1122.35047
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: The author studies generalized solutions of the initial value problem \[ \begin{gathered} u_t= u_{xx}+ f(u),\quad 0< t< T,\quad x\in\mathbb{R},\\ u|_{t=0}= u_0(x),\quad x\in\mathbb{R},\end{gathered}\tag{1} \] where \(f\) is expressed in the form \[ f(u)= g(u)- g(1) H(a-2)\tag{2} \] with \(\alpha\in (0,1)\) is a constant, \(H\) is the function on \(\mathbb{R}\) given by \[ H(u)= 0\quad\text{in }(-\infty, 0),\quad H(u)= 1\quad\text{in }(0,\infty)\quad\text{and }0\leq H(0)\leq 1\tag{3} \] and \(g\) is Lipschitz continuous function such that \(g(0)= 0\), \(g(u)< 0\) on \((0,\alpha]\) and \(g(u)- g(1)> 0\) on \([\alpha,1)\). The goal of this paper is to discuss the relationship between generalized and weak solutions for (1)--(3).
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5072968
zbMATH Keywords: parabolic initial value problems
zbMATH Keywords: discontinuous nonlinearities
zbMATH Keywords: generalized solutions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.10.052
OpenAlex ID: W1978750016
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DOI: 10.1016/J.JMAA.2005.10.052
description: scientific article; zbMATH DE number 5072968