Fatou theorem for the solutions of some nonlinear equations
type: Item
instance of: scholarly article
title: Fatou theorem for the solutions of some nonlinear equations
zbMATH Open document ID: 0815.35040
author: Kin Ming Hui
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-06-27T00:00:00Z
review text: The paper deals with the equation (1) \(u_ t = \Delta \Phi (u)\) where \(\Phi\) is continuous, strictly increasing in the interval \([0, \infty)\) and satisfies certain growth conditions, such that \(\Phi (0) = 0\), \(\Phi (u) > 0\) for \(u > 0\). Equation (1) generalizes the porous medium and fast diffusions equations. Assuming \(u(x,t)\) to be a nonnegative continuous weak solution of (1) \((x \in \mathbb{R}^ n\), \(0 < t < T)\) the author proves several theorems concerning the limit of \(u(\cdot,t)\) when \(t \to 0 +\). The paper generalizes the results concerning the porous medium equation given in [\textit{B. E. J. Dahlberg}, \textit{E. B. Fabes} and \textit{C. E. Kenig}, Proc. Am. Math. Soc. 91, 205-212 (1984; Zbl 0552.35047)].
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 599933
zbMATH Keywords: Fatou theorem
zbMATH Keywords: porous medium equation
zbMATH Keywords: weak limit
zbMATH Keywords: fast diffusions equations
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1129
OpenAlex ID: W1987893561
DOI: 10.1006/JMAA.1994.1129
description: scientific article; zbMATH DE number 599933