Existence and stability of multidimensional travelling waves in the monostable case
type: Item
instance of: scholarly article
title: Existence and stability of multidimensional travelling waves in the monostable case
zbMATH Open document ID: 0929.35064
DOI: 10.1007/BF02808184
author: A. I. Volpert
author: Vitaly Volpert
published in: Israel Journal of Mathematics
publication date: 1999-12-01T00:00:00Z
review text: The paper deals with the coupled system \[ \frac{\partial u}{\partial t}=a(x')\Delta u+\sum_{j=1}^m b_j(x') \frac{\partial u}{\partial x_j}+F(u, x') \] in a cylinder \(\Omega=\Omega'\times \mathbb{R}\). Here \[ u=u(x,t)=(u_1(x,t), \dots, u_n(x,t)), \quad x=(x_1, \dots, x_m), \quad x'=(x_2, \dots, x_m), \] \(x_1\) is the variable along the axis of the cylinder, \(\Omega'\) is a bounded domain. In addition \(F=(F_j(x))\) is a smooth vector-valued function defined on \(\mathbb{R}^n\times \Omega'\); \(a, b_j\) are diagonal matrices. The travelling wave solution \(w\) is a solution of the form \(u(x,t)=w(x_1-ct, x_2, \dots,x_m)\) with a constant \(c\). Under consideration, existence and stability conditions for travelling waves are derived provided the condition \[ \frac{\partial F_i}{\partial u_j}\geq 0, \quad i\neq j; \quad i, j=1, \dots, n, \] holds.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K40
zbMATH DE Number: 1310641
zbMATH Keywords: quasilinear parabolic systems
zbMATH Keywords: stability conditions
reviewed by: Michael I. Gil'
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02808184
OpenAlex ID: W1986707225
description: scientific article; zbMATH DE number 1310641