Asymptotic behaviour of solutions of \(u_t=\Delta\log u\) in a bounded domain
type: Item
instance of: scholarly article
title: Asymptotic behaviour of solutions of \(u_t=\Delta\log u\) in a bounded domain
zbMATH Open document ID: 1021.35055
author: Kin Ming Hui
published in: Differential and Integral Equations
publication date: 2003-03-13T00:00:00Z
review text: The author studies the large time behaviour of solutions of the equation \[ \frac{\partial u}{\partial t}=\Delta\log u,\quad u>0\tag{1} \] in a bounded cylindrical domain \(\Omega\times (0,\infty)\) with either finite or infinite lateral boundary values on \(\partial\Omega\times (0,\infty)\). The author shows that if \(u\) is the solution of (1), \(u=c_1\) on \(\partial\Omega\times (0,\infty)\), \(u(x,0)=u_0(x)\geq 0\) on \(\Omega\subset \mathbb{R}^n\), where \(\Omega\) is a smooth convex bounded domain, then for \(c_1=\infty\) the rescaled function \(w=\log(\frac ut)\) will converge uniformly on every compact subset of \(\Omega\) to the unique solution \(\psi\) of the equation \[ \Delta\psi-e^\psi=0,\quad \psi>0\text{ in }\Omega \] with \(\psi=\infty\) on \(\partial\Omega\) as \(t\to\infty\). Moreover he analyzes the case of finite \(c_1\).
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1832813
zbMATH Keywords: large time behaviour
zbMATH Keywords: finite or infinite lateral boundary values
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1832813