Self-similar solutions of the porous medium equation in a half-space with a nonlinear boundary condition: existence and symmetry
type: Item
instance of: scholarly article
title: Self-similar solutions of the porous medium equation in a half-space with a nonlinear boundary condition: existence and symmetry
zbMATH Open document ID: 1113.35077
author: Juan Dávila
author: J. D. Rossi
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-19T00:00:00Z
review text: In this work, it is proved that the problem \[ \begin{cases} \Delta u = u^{\alpha} &\text{in}\quad \mathbb R^N_+\\ \frac{\partial u}{\partial\nu} = u &\text{on}\quad \partial\mathbb R^N_+\end{cases} \] where \(\partial/\partial\nu\) is the outer unit normal derivative and \(0<\alpha<1\), has a nonnegative solution with compact support. It is proved for this problem that the support of every nonnegative solution with a finite energy is compact and radially symmetric in tangential variables.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2092952
zbMATH Keywords: Elliptic equations
zbMATH Keywords: Nonlinear boundary conditions
zbMATH Keywords: Symmetry
reviewed by: Valeriy Yumaguzhin
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.031
OpenAlex ID: W1985347991
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
cites work: Dual variational methods in critical point theory and applications
cites work: Blowup in diffusion equations: A survey
cites work: Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R}_+^n\) with a nonlinear boundary condition
cites work: On the blow-up set for u_t=du^m+u^m, m>1
cites work: Uniqueness and stability of regional blow-up in a porous-medium equation
cites work: Symmetry in an elliptic problem and the blow—up set of
cites work: On a semilinear elliptic problem in \(\mathbb{R}^ N\) with a non-Lipschitzian nonlinearity
cites work: On a blow-up set for the quasilinear heat equation \(u_ t=(u^ \sigma u_ x)_ x + u^{\sigma+1}\)
cites work: Continuation of blowup solutions of nonlinear heat equations in several space dimensions
cites work: The problem of blow-up in nonlinear parabolic equations
cites work: Symmetry and related properties via the maximum principle
cites work: Nondegeneracy of blowup for semilinear heat equations
cites work: Q3736159
cites work: Q4040640
cites work: Symmetry of the blow‐up set of a porous medium type equation
cites work: Symmetry results for reaction-diffusion equations
cites work: Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld
cites work: Q4035297
DOI: 10.1016/J.JMAA.2004.04.031
description: scientific article; zbMATH DE number 2092952