Traveling wave solutions for a nonlocal phase-field system
type: Item
instance of: scholarly article
title: Traveling wave solutions for a nonlocal phase-field system
zbMATH Open document ID: 1010.35107
DOI: 10.4171/IFB/59
author: Peter W. Bates
author: Fengxin Chen
published in: Interfaces and Free Boundaries
publication date: 2003-01-19T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: The authors study a nonlocal phase-field system which couples the temperature \(\theta \) and an order parameter \(\varphi \) according to the evolution equations: \[ \partial \varphi /\partial t=J*\varphi -\varphi +f\left( \varphi \right) +\theta, \qquad \partial \left( \theta +l\varphi \right) /\partial t=\Delta \theta. \] They first justify this model giving the expression of an associated free energy. The existence and uniqueness results have already been proved in a previous work by the authors in \textit{P.W. Bates} (ed.) et al. [Proceedings of the US-Chinese conference: Differential equations and applications, Hangzhou, China, 1996. Cambridge, MA: International Press, 14--21 (1997; Zbl 0938.35035)]. In the present paper, the authors prove the existence of traveling wave solutions, with a positive constant and unknown speed \(c\), thus restricting the problem to the one-dimensional case. Their main theorem proves the existence of such traveling waves provided that the parameter \(l\), which represents the latent temperature, is small enough. Solving the second above equation in this framework and taking \(l=0\), the authors first prove the existence of solutions \(\left( \varphi _0,c_0\right) \) of the first equation which can be written as : \(c\varphi '+J*\varphi -\varphi +f\left( \varphi \right) +\theta \left( -\infty \right) \), where \(\varphi _0\) is smooth if \(c_0\) is positive. Then they study the linear operator which associates to every continuously differentiable function \(\psi \) vanishing at \(\pm \infty \) together with its first-order derivative, the function \(c\psi '+J*\psi -\psi +f'\left( \varphi _0\right) \psi \). The proof is then based on Fredholm theory and the implicit function theorem. The end of the paper presents some numerical results for different expressions of the function \(f\), which is supposed to be bistable, and different values of the parameter \(l\).
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 1791238
zbMATH Keywords: nonlocal phase transition
zbMATH Keywords: existence of traveling wave solutions
zbMATH Keywords: continuation method
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1791238