On positive solutions of the heat equation satisfying a nonlinear boundary condition
type: Item
instance of: scholarly article
title: On positive solutions of the heat equation satisfying a nonlinear boundary condition
zbMATH Open document ID: 1209.35056
DOI: 10.1134/S0012266110080057
published in: Differential Equations
publication date: 2010-11-29T00:00:00Z
review text: The following inequality is considered: \[ \begin{alignedat}{2} {\partial u\over \partial t} &\geq \Delta u&\quad& \text{in } \Omega\times (0,\infty),\\ {\partial u\over \partial {\mathbf n}} &\geq {u^{\sigma}\over (1+t)^{\alpha}}&\quad & \text{on } \partial \Omega\times (0,\infty).\end{alignedat} \] Here, \(\Omega\) is an unbounded domain in \(\mathbb{R}^n\), \(\partial/\partial {\mathbf n}\) is the directional derivative along the outward normal to \(\partial \Omega\), and \(\sigma>1\) and \(\alpha\in \mathbb{R}\) are constants. For each dimension \(n\), the paper provides sharp conditions on the exponents \(\sigma\) and \(\alpha\) for which the above inequality has a classical positive solution. Note that the nonexistence of such a solution implies the nonexistence of a global solution to the corresponding heat equation because the blow-up of the solution takes place.
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 5820359
zbMATH Keywords: classical solution
author: V. A. Kondrat'ev
MaRDI profile type: Publication
Wikidata QID: Q115252502
cites work: On nonexistence of global solutions of some semilinear parabolic differential equations
cites work: The blow‐up rate for the heat equation with a non‐linear boundary condition
cites work: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary
cites work: On critical exponents for the heat equation with a nonlinear boundary condition
full work available at URL: s0012266110080057
OpenAlex ID: W2071954977
description: scientific article; zbMATH DE number 5820359