Parabolic problems with nonlinear boundary conditions and critical nonlinearities
type: Item
instance of: scholarly article
title: Parabolic problems with nonlinear boundary conditions and critical nonlinearities
zbMATH Open document ID: 0938.35077
author: Jose M. Arrieta
author: A. N. Carvalho
author: Anibal Rodriguez-Bernal
published in: Journal of Differential Equations
publication date: 2000-06-12T00:00:00Z
review text: The authors study the nonlinear parabolic problem \[ \begin{gathered} \frac {\partial u}{\partial t} =\Delta u + f(u) \text{ in } \Omega\times(0,T), \\ \frac {\partial u}{\partial \nu} = g(u) \text{ on } \partial\Omega\times(0,T), \\ u(\cdot,0) = u_0 \text{ in } \Omega, \end{gathered} \] where \(\Omega \subset \mathbb R^N\) has unit outer normal \(\nu\), and \(u\) depends on \(x\) and \(t\). Their main theorem is that if there are constants \(q \in (1,\infty)\) and \(C\) such that \[ \begin{aligned} |f(u)-f(v)|&\leq C|u-v|(|u|^{2q/N}+|v|^{2q/N} +1),\\ |g(u)-g(v)|&\leq C|u-v|(|u|^{q/N}+|v|^{q/N} +1) \end{aligned} \tag{*} \] (if \(N \geq 2\)), and if \(u_0 \in L^q(\Omega)\), then this problem has a unique solution. (If \(N=1\), the exponents in the inequality for \(g\) must be less than \(q\).) They also show that if \(u_0 \in W^{1,q}\) with \(q \in(1,\infty)\), then the condition (*) can be relaxed. These results include all known results on existence for problems with critical growth. The method of proof relies on various interpolation theorems for scales of Banach spaces.
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1339375
zbMATH Keywords: interpolation theorems for scales of Banach spaces
zbMATH Keywords: critical growth
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
OpenAlex ID: W1972186223
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DOI: 10.1006/JDEQ.1998.3612
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description: scientific article; zbMATH DE number 1339375