On elliptic equations related to self-similar solutions for nonlinear heat equations
type: Item
instance of: scholarly article
title: On elliptic equations related to self-similar solutions for nonlinear heat equations
zbMATH Open document ID: 0626.35033
author: Yoshikazu Giga
published in: Hiroshima Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The author studies the nonlinear elliptic equation \[ (*)\quad \Delta w- ()x\cdot \nabla w-\alpha w+| w|^{p-1} w=0\quad in\quad {\mathbb{R}}^ n, \] where \(p>1\) is fixed and \(\alpha\geq 0\) is a parameter. This equation is obtained from the nonlinear heat equation \(u_ t-\Delta u-| u|^{p-1} u=0\) by introducing a suitable similarity space variable and separating the time variable. Only global positive solutions are considered which are radial in the sense that they depend on \(r=| x|\) alone. A constant positive solution only exists for \(\alpha =1/(p-1)\). It turns out that this value is a ``bifurcation value'' of the parameter \(\alpha\), the precise meaning of which is given by the following main content of the paper: (a) If \(\alpha >1/(p-1)\) and \(n<2(p+1)/(p-1)\), then there exists a positive radially decreasing global solution w(r) of (*). (b) If \(\alpha\leq 1/(p-1)\) and \(n\leq 2(p+1)/(p-1)\), no such solution exists. Moreover, in case (a) the asymptotic decay of w(r) is not exponential but of finite order with respect to r. The elegant proofs make use of the shooting method, the variational method, and a certain integral identity.
reviewed by: Josef Hainzl
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4017461
zbMATH Keywords: similarity space variable
zbMATH Keywords: separating the time variable
zbMATH Keywords: global positive solutions
zbMATH Keywords: radial
zbMATH Keywords: bifurcation value
zbMATH Keywords: asymptotic decay
zbMATH Keywords: shooting method
zbMATH Keywords: variational method
zbMATH Keywords: integral identity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4017461