An entire solution of a bistable parabolic equation on \(\mathbb{R}\) with two colliding pulses
type: Item
instance of: scholarly article
title: An entire solution of a bistable parabolic equation on \(\mathbb{R}\) with two colliding pulses
zbMATH Open document ID: 1362.35152
published in: Journal of Functional Analysis
publication date: 2017-01-17T00:00:00Z
review text: The authors are concerned with the study of the parabolic equation \[ u_t=u_{xx}+f(u), \quad x\in {\mathbb R}, \, t\in {\mathbb R}.\eqno{(*)} \] The function \(f\in C^1({\mathbb R})\) satisfies: there exist \(\gamma>\beta>0\) such that \(f(0)=f(\beta)=f(\gamma)=0\), \(f'(0)<0\), \(f'(\beta)>0\), \(f'(\gamma)<0\), \(f<0\) in \((0,\beta)\), \(f>0\) in \((\beta,\gamma)\) and \(\int_0^\gamma f(s)ds>0\). The main result of the paper reads as follows. There exists a solution \(U\) of (*) with the following properties: (i) \(0<U(\cdot,t)<\gamma\), \(U(\cdot,t)\in C_0({\mathbb R})\) and \(U(\cdot,t)\) is even, for all \(t\in {\mathbb R}\), (ii) For each \(t<0\), \(U(\cdot,t)\) has exactly three critical points, precisely two global maximum points and one local minimum point at \(x=0\). For all sufficiently large \(t>0\), \(x=0\) is the unique critical point of \(U(\cdot,t)\) and it is the global maximum point of \(U(\cdot,t)\), (iii) \(\lim_{t\to \infty}\|U(\cdot,t)-\phi\|_{L^\infty({\mathbb R})}=0\), where \(\phi\) satisfies \(\phi_{xx}+f(\phi)=0\) in \({\mathbb R}\) and \(\phi'(0)=0\), (iv) There exists a \(C^1\) function \(\zeta\in C^1(-\infty,0)\) such that \(\zeta(t)\to \infty\), \(\zeta'(t)\to 0\) as \(t\to -\infty\) and \[ \lim_{t\to -\infty}\|U(\cdot,t)-\phi(\cdot-\zeta(t))-\phi(\cdot+\zeta(t))\|_{L^\infty({\mathbb R})}=0. \]
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B08
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 6675456
zbMATH Keywords: parabolic equations on \(\mathbb{R}\)
zbMATH Keywords: entire solutions
zbMATH Keywords: slowly moving pulses
zbMATH Keywords: threshold solutions
author: Hiroshi Matano
author: Peter Poláčik
MaRDI profile type: Publication
full work available at URL: j.jfa.2016.11.006
OpenAlex ID: W2554034365
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DOI: 10.1016/J.JFA.2016.11.006
description: scientific article; zbMATH DE number 6675456