On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: Intermediate solutions
type: Item
instance of: scholarly article
title: On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: Intermediate solutions
zbMATH Open document ID: 0946.35007
DOI: 10.1215/S0012-7094-98-09424-8
author: Wei-Ming Ni
author: Izumi Takagi
author: Juncheng Wei
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: This article continues investigations of \textit{W.-M. Ni} and \textit{J. Wei} [Commun. Pure Appl. Math. 48, No. 7, 731-768 (1995; Zbl 0838.35009)]. The authors consider the problem \[ \begin{cases}\varepsilon^2\Delta u+f(u)=0 \;&\text{ in } G,\\ u>0 \;&\text{ in } G,\\ u=0 \;&\text{ on } \partial G, \end{cases} \] \(G\) is a smooth bounded domain in \(\mathbb{R}^n\). A typical example for the possible function \(f\) is the cubic function \(f(u)=u(u-a)(1-u)\) with \(0<a<1/2\). The location of spikes as well as the profile of spikes are established. The basic idea in proving such results is a combination of the vanishing viscosity method and the energy method.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1425150
zbMATH Keywords: spike-layer solutions
zbMATH Keywords: intermediate solutions
zbMATH Keywords: singular perturbation
zbMATH Keywords: semilinear Dirichlet problem
reviewed by: E. A. Kalita
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09424-8
OpenAlex ID: W1579246209
description: scientific article; zbMATH DE number 1425150