Some remarks on the stability of sign changing solutions
type: Item
instance of: scholarly article
title: Some remarks on the stability of sign changing solutions
zbMATH Open document ID: 0837.35020
author: Zongming Guo
author: E. N. Dancer
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-05-19T00:00:00Z
review text: This paper deals with the study of the stability properties of the changing sign solutions of the problems: \[ - \Delta u= h(u)\quad\text{in}\quad D,\quad u= 0\quad\text{on}\quad \partial D,\quad\text{or}\quad {\partial u\over \partial n}= 0\quad\text{on}\quad \partial D, \] \[ - \varepsilon^2 \Delta u= h(u)\quad\text{in}\quad D,\quad u= 0\quad\text{on}\quad \partial D,\quad\text{or}\quad {\partial u\over \partial n}= 0\quad\text{on}\quad \partial D, \] where \(D\) is a bounded domain in \(\mathbb{R}^n\) \((n\geq 2)\) with regular boundary, \(\varepsilon> 0\) and \(h: \mathbb{R}^1\to \mathbb{R}^1\) is defined by \[ h(u)= \begin{cases} \alpha u- \alpha u^2\quad & \text{if}\quad u\geq 0\\ du+ u^2\quad & \text{if} \quad u\leq 0\end{cases} \] where \(\alpha> 0\). By stability is meant stability for the natural corresponding parabolic problem. The instability of many sign changing solutions is proved. A number of methods are found for obtaining stable changing sign solutions. Some of these methods involve singular perturbations.
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 794314
zbMATH Keywords: instability
zbMATH Keywords: singular perturbations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225592
OpenAlex ID: W1970004407
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DOI: 10.2748/TMJ/1178225592
description: scientific article; zbMATH DE number 794314