Front propagation in nonlinear parabolic equations
type: Item
instance of: scholarly article
title: Front propagation in nonlinear parabolic equations
zbMATH Open document ID: 1315.35018
author: Eduard Feireisl
author: Hana Petzeltová
author: Peter Takáč
author: Danielle Hilhorst
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.0379
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35C07
zbMATH DE Number: 6355584
zbMATH Keywords: nonlinear parabolic equations
zbMATH Keywords: interface
zbMATH Keywords: front speed
zbMATH Keywords: travelling waves
Wikidata QID: Q59316483
MaRDI profile type: Publication
OpenAlex ID: W2033009995
arXiv ID: 1309.0379
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DOI: 10.1112/JLMS/JDU039
published in: Journal of the London Mathematical Society. Second Series
review text: This paper deals with the following Cauchy problem (P) for an unknown function \(u(x,t)\): NEWLINE\[NEWLINE\partial_t u=\text{div}(\partial \Phi(\nabla u))+f(u),\quad u(x,0)=u_o(x),\quad x\in\mathbb R^N, t>0,NEWLINE\]NEWLINE where \(\Phi:\mathbb R^N\to \mathbb R, f:\mathbb R\to \mathbb R, u_o:\mathbb R^N\to\mathbb R\) are suitable given data and \(\partial \Phi\) is the Fréchet derivative of \(\Phi\). In particular, it is assumed that \(\Phi\) is radially symmetric and that there is a constant \(\mu\in (-1,1)\) such that \(f\in C(\mathbb R),f(-1)=f(\mu)=f(1)=0, f<0\text{ in }(-1,\mu),f>0 \text{ in } (\mu,1)\). A standard change of variables leads to the scaled problem \((P_{\epsilon})\): NEWLINE\[NEWLINE\partial_t u_{\epsilon}=\text{div}(\partial \Phi(\epsilon\nabla u_{\epsilon}))+\frac{1}{\epsilon}f(u_{\epsilon}),\quad u_{\epsilon}(x,0)=u_{\epsilon,o}(x),\quad x\in\mathbb R^N, t>0.NEWLINE\]NEWLINE The authors investigate the behavior of solutions \(u_{\epsilon}\) to problem \((P_\epsilon)\) as \(\epsilon\to 0.\) When \(N=1\), setting \(u(x,t)=q(x-ct)\) in (P), where \(c\) is the speed of the wave, leads to the associated one dimensional problem (Q): NEWLINE\[NEWLINEd_x(\partial\Phi (q^{\prime}))+cq^{\prime}+f(q)=0,\quad x\in \mathbb RNEWLINE\]NEWLINE with the conditions NEWLINE\[NEWLINEq^{\prime}\leq 0 \text{ in }\mathbb R,\quad \underset{x\to -\infty}{\lim}q(x)=1,\quad \underset{x\to \infty}{\lim}q(x)=-1, \text{ and } q(0)=\mu.NEWLINE\]NEWLINE They give additional conditions on \(f\) and \(\Phi\) which imply the uniqueness of the solution \((q,c)\) to (Q), such that: NEWLINE\[NEWLINEc=(\int_{\infty}^{\infty}|q^{\prime}|^2 dx)^{-1} \int_{-1}^{1}f(s)ds\geq 0,NEWLINE\]NEWLINE is the set of all. They introduce two regions \(G_{-}\) [resp. \(G_{+}\)] \(x\in\mathbb R^N\) such that there exists a neighbourhood \(U(x)\subset\mathbb R^N\) of \(x\) such that \(\underset{\epsilon\to 0}{\limsup}(\underset{y \in U(x)}{\sup} u_{\epsilon,o}(y))<\mu\), [resp. \(\underset{\epsilon…
reviewed by: Denise Huet
description: scientific article; zbMATH DE number 6355584