Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov's equation
type: Item
instance of: scholarly article
title: Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov's equation
zbMATH Open document ID: 1097.34030
author: Juan Campos
author: Aureliano M. Robles-Pérez
author: Margarita Arias
author: Luis Sanchez
published in: Calculus of Variations and Partial Differential Equations
publication date: 2004-12-17T00:00:00Z
review text: It is well known that travelling wave solutions with speed \(c\) of the Fisher-Kolmogorov equation \(u_t= u_{xx}+ f(u)\), where \(f: [0,1]\to \mathbb{R}^+\) is Lipschitzian and satisfies \(f(0)= 0= f(1)\) and \(f(u)> 0\) for \(0< u< 1\), correspond to positive heteroclinic solutions of the second-order equation \(u''+ cu'+ f(u)= 0\). It is well known that such solutions exist for \(c\geq c^*> 0\). The authors consider the more general equation \[ u''+ cu'+\lambda f(u)= 0\tag{\(*\)} \] and call a heteroclinic solution \(u\) of \((*)\) for some \(\lambda> 0\) as fast heteroclinic solution if it belongs to the space \(H^1_{\text{loc}}(\mathbb{R}): e^{ct} u\in H^1(\mathbb{R})\) and satisfies \(u(-\infty)= 1\). They give a variational characterization of fast heteroclinic orbits as well as of \(c^*\).
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 2123515
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: s00526-004-0264-y
OpenAlex ID: W2077701732
DOI: 10.1007/S00526-004-0264-Y
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description: scientific article; zbMATH DE number 2123515