Large-time behavior of small solutions of a two-dimensional semilinear elliptic equation with a dynamical boundary condition

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: -Δu=up, x∈R2+, t>0, ∂tu+∂νu=0, x∈∂R2 +, t>0, u(x,0)=φ(x1)≥0, x=(x1,0) ∈∂R2+, where u=u(x,t), ∂t:= ∂/∂t, ∂ν:=-∂/∂x2, R 2+:={(x1,x2): x1∈R, x2>0} and p>1. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel. This is an extension of previous results of the authors of this paper to the two-dimensional case.

Original languageEnglish
Pages (from-to)107-123
Number of pages17
JournalAsymptotic Analysis
Volume85
Issue number1-2
DOIs
Publication statusPublished - 2013 Dec 4

Keywords

  • Dynamical boundary condition
  • Large time behavior
  • Semilinear elliptic equation in a half-plane

ASJC Scopus subject areas

  • Mathematics(all)

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