On the Cauchy problem for the equation \(u'-\Delta u=u| u| ^{\gamma -1}\) in the supercritical case
type: Item
instance of: scholarly article
title: On the Cauchy problem for the equation \(u'-\Delta u=u| u| ^{\gamma -1}\) in the supercritical case
zbMATH Open document ID: 0681.35043
DOI: 10.1007/BF03323237
author: Wilfried Wieser
published in: Results in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: We prove global existence of classical solutions to the above problem whenever the initial function is continuous and can be enclosed between a sub- and a supersolution of the corresponding stationary problem, where the latter are assumed to be smooth bounded functions lying in some \(L^ p\). (In the title, ``supercritical'' means \(\gamma \geq (n+2)/(n-2)\) where n is space dimension.) The method of proof combines local existence results from semilinear parabolic theory with results on the corresponding stationary equation via the use of the Phragmén-Lindelöf principle for parabolic equations in unbounded domains. Examples are provided, illustrating the extension of previously known global existence results.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 4115131
zbMATH Keywords: semilinear
zbMATH Keywords: global existence
zbMATH Keywords: critical exponent
zbMATH Keywords: Phragmén-Lindelöf principle
zbMATH Keywords: Cauchy problem
zbMATH Keywords: corresponding stationary equation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03323237
OpenAlex ID: W1999272674
description: scientific article; zbMATH DE number 4115131