A priori \(L^\infty\)-estimate and existence of solutions for some nonlinear parabolic equations
type: Item
instance of: scholarly article
title: A priori \(L^\infty\)-estimate and existence of solutions for some nonlinear parabolic equations
zbMATH Open document ID: 0957.35028
DOI: 10.1016/S0362-546X(99)00135-2
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-03-16T00:00:00Z
review text: The author studies the following nonlinear parabolic problem \[ \begin{cases} u_t-\text{ div}(a(x,t,u,\nabla u)+\Phi(u)) =b(x,t,u,\nabla u) -\text{div} g,\\ u|_\Sigma =0,\cr u(x,0)=u_0(x)\in L^\infty(\Omega),\end{cases} \] where \(\Omega\) is a bounded domain in \({\mathbb R}^N\), \(\Sigma=\partial \Omega\times(0,T).\) The functions \(a\) and \(b\) are supposed to be Carathéodory functions satisfying suitable coercivity, growth and monotonicity conditions, respectively. It is proved: an \(L^\infty\)-a priori estimate for the solution of the problem and its weak solvability in the space \(L^\infty(Q)\cap L^p(0,T;W^{1,p}_0(\Omega))\cap C([0,T], L^q(\Omega))\) for all \(q\in[1,\infty).\)
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1513301
zbMATH Keywords: Carathéodory functions
zbMATH Keywords: coercivity, growth and monotonicity conditions
zbMATH Keywords: natural growth
author: Shulin Zhou
reviewed by: Lubomira Softova
MaRDI profile type: Publication
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full work available at URL: s0362-546x(99)00135-2
OpenAlex ID: W2094555801
description: scientific article; zbMATH DE number 1513301