Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary
type: Item
instance of: scholarly article
title: Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary
zbMATH Open document ID: 1390.35119
author: Yuki Kaneko
author: Hiroshi Matsuzawa
published in: Journal of Differential Equations
publication date: 2018-04-24T00:00:00Z
review text: The paper is concerned with the study of the free boundary problem \(u_t=u_{xx}+f(u)\), \(t>0\), \(0<x<h(t)\) subject to the following conditions: \(u(t,ct)=u(t,h(t))=0\), \(h'(t)=-\mu u_x(t, h(t))\) for \(t>0\) and \(h(0)=h_0\), \(u(0,x)=u_0(x)\), \(0\leq x\leq h_0\). Here \(c,\mu, h_0>0\) are given constants and \(f\in C^1\) satisfies \(f(0)=f(1)=0\), \(f'(1)<0\), \(f(u)<0\) for \(u>1\). The dynamical behavior of solutions to the above problem is considered in the frame of three type on extra-assumptions on \(f\): monostable case; bistable case; combustion case. It is derived that the unique solution exhibits exactly one of the behaviors: spreading, vanishing and transition. The asymptotic profile of the solution over the whole domain when spreading occurs.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 6864107
zbMATH Keywords: free boundary problem
zbMATH Keywords: nonlinear diffusion equation
zbMATH Keywords: monostable
zbMATH Keywords: bistable
zbMATH Keywords: combustion
MaRDI profile type: Publication
full work available at URL: j.jde.2018.03.026
OpenAlex ID: W2800369554
Wikidata QID: Q111164040
cites work: The zero set of a solution of a parabolic equation.
cites work: Q4089232
cites work: Multidimensional nonlinear diffusion arising in population genetics
cites work: Asymptotic behavior of solutions of Fisher-KPP equation with free boundary conditions
cites work: Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions
cites work: Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary
cites work: Spreading and vanishing in nonlinear diffusion problems with free boundaries
cites work: Nonlinear Diffusion Problems with Free Boundaries: Convergence, Transition Speed, and Zero Number Arguments
cites work: Convergence and sharp thresholds for propagation in nonlinear diffusion problems
cites work: Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems
cites work: Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary
cites work: Spreading in advective environment modeled by a reaction diffusion equation with free boundaries
cites work: Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries
cites work: Remarks on Spreading and Vanishing for Free Boundary Problems of Some Reaction-Diffusion Equations
cites work: Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations
cites work: A free boundary problem for a reaction-diffusion equation appearing in ecology
cites work: Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity
cites work: Q5562267
cites work: Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model
cites work: Q4354424
cites work: Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions
cites work: On a reaction-diffusion equation with Robin and free boundary conditions
cites work: A free boundary problem for the Fisher-KPP equation with a given moving boundary
cites work: Q4765401
cites work: RANDOM DISPERSAL IN THEORETICAL POPULATIONS
DOI: 10.1016/J.JDE.2018.03.026
description: scientific article; zbMATH DE number 6864107