When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?
type: Item
instance of: scholarly article
title: When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?
zbMATH Open document ID: 0939.35014
DOI: 10.1007/BF02791135
author: Shigeru Sakaguchi
published in: Journal d'Analyse Mathématique
publication date: 2000-06-26T00:00:00Z
review text: The author considers heat equations on a bounded domain under the Neumann boundary condition. He classifies the solutions whose level surfaces are invariant with respect to the time variable. He extends earlier results of Alessandrini who studied the problem for the Dirichlet boundary condition. In the latter case, the invariance is possible only if the solution is radially symmetric (and the domain is a ball) or if it coincides with an eigenfunction of the Laplace operator. In the Neumann case, there are additional possibilities. Also there are extra difficulties to be dealt with. The author also addresses the classification problem for some nonlinear diffusion equations, such as the porous medium equation.
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 1369962
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: porous medium equation
zbMATH Keywords: invariance of level sets
zbMATH Keywords: symmetry
reviewed by: Peter Poláčik
MaRDI profile type: Publication
cites work: Matzoh ball soup: A symmetry result for the heat equation
cites work: Characterizing spheres by functional relations on solutions of elliptic and parabolic equations
cites work: Stabilization of solutions of the filtration equation with absorption and nonlinear flux
cites work: The zero set of a solution of a parabolic equation.
cites work: Solutions of the One-Dimensional Porous Medium Equation are Determined by Their Free Boundary
cites work: Large time behaviour of solutions of the porous medium equation in bounded domains
cites work: Q3940162
cites work: Stability of the separable solution for fast diffusion
cites work: A positivity property of solutions of nonlinear diffusion equations
cites work: The Initial-Neumann Problem for the Heat Equation in Lipschitz Cylinders
cites work: Q4325977
cites work: Q5520962
cites work: Q3245454
cites work: Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems
cites work: Movement of hot spots over unbounded domains in \({\mathbb{R}}^ N\)
cites work: Q5562267
cites work: Q5767686
cites work: The Number of Peaks of Positive Solutions of Semilinear Parabolic Equations
cites work: Q4080368
cites work: Critical point theory and submanifold geometry
cites work: An introduction to partial differential equations
cites work: Continuity of solutions of a singular parabolic equation
cites work: Behavior Near t=0 for Solutions of the Dirichlet Problem foru(t)= δφ(u)−f(u)in Bounded Domains
cites work: Q5770505
cites work: A symmetry problem in potential theory
cites work: Q3782002
cites work: Weakly Differentiable Functions
full work available at URL: https://doi.org/10.1007/bf02791135
OpenAlex ID: W2054490728
description: scientific article; zbMATH DE number 1369962