Spreading and vanishing in nonlinear diffusion problems with free boundaries
type: Item
instance of: scholarly article
title: Spreading and vanishing in nonlinear diffusion problems with free boundaries
zbMATH Open document ID: 1331.35399
DOI: 10.4171/JEMS/568
author: Yihong Du
author: Bendong Lou
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2015-11-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.5373
review text: The authors study classical solutions to the free boundary problem \[ \begin{cases} u_t = u_{xx} + f(u), \qquad & (t,x) \in (0,\infty) \times (g(t),h(t)), \\ u(t,g(t))= u(t,h(t))=0, \qquad & t \in (0,\infty), \\ g^\prime (t) = -\mu u_x(t,g(t)), \; h^\prime (t) = -\mu u_x(t,h(t)), \qquad & t \in (0,\infty), \\ -g(0) = h(0) = h_0, \; u(0,x) = u_0(x), \qquad & x \in [-h_0, h_0], \end{cases} \] for the semilinear heat equation, where \(\mu\) and \(h_0\) are positive, \(f \in C^1([0,\infty))\) with \(f(0)=0\), and the initial data \(u_0 \in C^2 ([-h_0,h_0])\) are positive and satisfy the Dirichlet boundary conditions along with \(u_0^\prime \neq 0\) on the boundary. The authors prove that any global bounded solution \(u\) converges to a single stationary solution. For \(f\) being of monostable, bistable or combustion type, the authors further classify the large time behavior of the solution and distinguish, in particular, between spreading and vanishing (i.e., \(u(t)\) converges to 1 or 0 as \(t \to \infty\), respectively). In the case of spreading, they further determine the asymptotic speed of the wave front described by the free boundary. Important ingredients of their proofs are semi-waves and waves of finite length which are used as sub- or supersolutions and allow to adapt the concept of traveling waves to the free boundary problem.
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6509929
zbMATH Keywords: nonlinear diffusion equation
zbMATH Keywords: free boundary problem
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: monostable
zbMATH Keywords: bistable
zbMATH Keywords: combustion
zbMATH Keywords: sharp threshold
zbMATH Keywords: spreading speed
MaRDI profile type: Publication
arXiv ID: 1301.5373
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OpenAlex ID: W3103452764
description: scientific article