The spatial critical points not moving along the heat flow
type: Item
instance of: scholarly article
title: The spatial critical points not moving along the heat flow
zbMATH Open document ID: 0892.35010
DOI: 10.1007/BF02788032
author: Rolando Magnanini
author: Shigeru Sakaguchi
published in: Journal d'Analyse Mathématique
publication date: 1997-07-23T00:00:00Z
review text: The authors consider the heat equation \(u_t= \Delta u\) in \(\mathbb{R}^N \times (0, \infty)\), \(u(x,0) =\varphi (x)\), and the spatial critical set of the solutions \(C(t)= \{x\in \mathbb{R}^N \mid \nabla u(t,x) =0\}\). The basic problem in this paper is the following: If the spatial critical point does not move in \(t\), then what can we say about the solution? The author proves a certain balance law on the initial condition \(\varphi\). That is if the origin 0 satisfies \(\nabla u(0,t) =0\) in some time interval \((a,b)\), then \(\int_{S^{N-1}} \omega\varphi (r\omega) d\omega =0\) holds for almost all \(r\geq 0\). The case of a bounded domain is also considered, and several kinds of balance laws and their relation to the symmetry of the situation are studied.
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 1038448
zbMATH Keywords: balance law
reviewed by: Shuichi Jimbo
MaRDI profile type: Publication
cites work: Solutions of the One-Dimensional Porous Medium Equation are Determined by Their Free Boundary
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cites work: Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems
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cites work: Movement of hot spots over unbounded domains in \({\mathbb{R}}^ N\)
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description: scientific article; zbMATH DE number 1038448