Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension
type: Item
instance of: scholarly article
title: Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension
zbMATH Open document ID: 1055.35065
author: Maria Gokieli
author: Frédérique Simondon
published in: Journal of Evolution Equations
publication date: 2004-09-22T00:00:00Z
review text: Consider a non-negative and bounded global weak solution \(u=u(t,x)\) to the (possibly) degenerate parabolic equation \(\partial_t u - \partial_x^2\varphi(u) = f(u)\) for \((t,x)\in (0,+\infty)\times (-L,L)\) with boundary conditions \(\pm\partial_x\varphi(t,\pm L) + \beta(\pm L)\;\varphi(u)(t,\pm L)=h(\pm L)\) and a non-negative initial datum \(u_0\in L^\infty(-L,L)\). Here \(L>0\), \(\varphi\in{\mathcal C}([0,+\infty))\cap {\mathcal C}^3((0,+\infty))\) and \(f\in {\mathcal C}^1([0,+\infty))\) satisfy \(\varphi(0)=f(0)=0\) and \(\varphi'(z)>0\) for \(z>0\), while \(\beta(\pm L)\) and \(h(\pm L)\) are four non-negative real numbers. It is shown that there is a stationary solution \(u_\infty\) such that \(u(t)\) converges uniformly in \([-L,L]\) towards \(u_\infty\) as \(t\to +\infty\). As \(\varphi'(0)\) may vanish, there might be a continuum of stationary solutions, and this is the main difficulty in the proof (if the set of stationary solutions is discrete, standard arguments from the theory of dynamical systems readily imply the result since a Lyapunov functional is associated to the above equation). This difficulty is overcome by a careful study of the set of stationary solutions and the use of the existing theory about the zero level set of solutions to one-dimensional parabolic equations.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 2101650
zbMATH Keywords: large time behaviour
zbMATH Keywords: continuum of stationary solutions
zbMATH Keywords: Lyapunov functional
reviewed by: Philippe Laurençot
MaRDI profile type: Publication
full work available at URL: s00028-003-0085-z
OpenAlex ID: W4240900984
DOI: 10.1007/S00028-003-0085-Z
description: scientific article; zbMATH DE number 2101650