A note on the paper by Qi S. Zhang: A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem
type: Item
instance of: scholarly article
title: A note on the paper by Qi S. Zhang: A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem
zbMATH Open document ID: 0954.35080
author: Philippe Souplet
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-01-02T00:00:00Z
review text: The author considers nonnegative solutions of the inhomogeneous heat equation \(u_t - \Delta u=f \geq 0\) on \(\mathbb{R}^n \times ] 0,T[\), with the initial condition that \(u(.,0)=u_0 \geq 0\) in the sense of local \(L^1\) convergence. Under certain smoothness conditions on \(f, u_0\) and \(u\), and assuming the existence of \(u\), he establishes the representation of \(u\) as the sum of the Gauss-Weierstrass integral of \(u_0\) and the heat potential of \(f\). Since \(u\) is a nonnegative supertemperature, the Riesz decomposition theorem plus the Widder representation theorem would almost have given the result. Among the closely related results which the author does not mention, perhaps the most general were obtained by the reviewer [J. Math. Anal. Appl. 67, 513-524 (1979; Zbl 0408.35045)]. For the paper mentioned in the title see ibid. 232, No. 2, 413-427, Art No. ID jmaa. 1999.6296 (1999; Zbl 0926.35058)].
reviewed by: Neil A. Watson
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 31B35
zbMATH DE Number: 1439373
zbMATH Keywords: heat potential
zbMATH Keywords: Riesz decomposition theorem
zbMATH Keywords: Gauss-Weierstrass integral
zbMATH Keywords: Widder representation theorem
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6648
OpenAlex ID: W2025254747
cites work: Existence and nonexistence of global solutions for \(u_ t = \Delta u + a(x)u^ p\) in \(\mathbb{R}^ d\)
cites work: Q3994731
cites work: Existence and non-existence of global solutions for a semilinear heat equation
cites work: Positive Temperatures on an Infinite Rod
cites work: A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem
DOI: 10.1006/JMAA.1999.6648
description: scientific article; zbMATH DE number 1439373