Large time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition
type: Item
instance of: scholarly article
title: Large time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition
zbMATH Open document ID: 1281.35051
author: Kazuhiro Ishige
author: Marek Fila
author: Tatsuki Kawakami
published in: Advances in Differential Equations
publication date: 2013-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ade/1355867482
review text: In this paper, the semilinear elliptic equation with a dynamical boundary condition \[ \begin{cases} -\Delta u = u^p, & x \in \mathbb{R}_+^N, \;t>0, \\ \partial_t u + \partial_\nu u = 0, & x \in \partial\mathbb{R}_+^N, \;t>0, \\ u(x,0)=\varphi(x') \geq 0, & x = (x',0) \in \partial\mathbb{R}_+^N \end{cases} \] is considered, where \(N \geq 3\), \(\mathbb{R}_+^N:=\{ x=(x',x_N) : x' \in \mathbb{R}^{N-1}, x_N>0 \}\), \(u=u(x,t)\), \(\Delta\) is the \(N\)-dimensional Laplacian, \(\partial_t:=\partial/\partial t\), \(\partial_\nu:=-\partial/\partial x_N\), and \(p>1\). It is shown that if \(p \leq (N+1)/(N-1)\), then there are no nontrivial global solutions, and that if \(p>(N+1)/(N-1)\), then there is a global minimal solution for a small \(\varphi\). Furthermore, the asymptotic behavior of the global minimal solution is also studied.
reviewed by: Satoshi Tanaka
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6141571
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6141571