A note on a nonautonomous O.D.E. related to the Fisher equation
type: Item
instance of: scholarly article
title: A note on a nonautonomous O.D.E. related to the Fisher equation
zbMATH Open document ID: 1035.34016
DOI: 10.1016/S0377-0427(99)00254-X
published in: Journal of Computational and Applied Mathematics
publication date: 2000-01-01T00:00:00Z
review text: Consider the nonautonomous differential equation \[ d^2u/dt^2 + c \;du/dt + a(t)f(u)=0, \tag \(*\) \] with \(c>0\) under the assumptions: (i) \(f: \mathbb{R}^+ \rightarrow \mathbb{R}^+\) is locally Lipschitzian and satisfies \(f(0)=0.\) (ii) \(a: \mathbb{R} \rightarrow \mathbb{R}\) is continuous. There is \(\alpha > 0\) such that \(\alpha < a (t) \leq 1\) for all \(t\). (iii) \(f(1)=0\), \(f(u) > 0\) for \(0 < u < 1.\) Under these assumptions, the author proves that \((*)\) has a positive solution \(u(t)\) satisfying \(u(- \infty ) =1\), \(u(+\alpha ) =0\), provided \(c\) satisfies some inequality. He establishes this result by comparing \((*)\) with some linear equations. By the same technique, the author proves the existence of a positive solution to \((*)\) satifying \(u(0)=u(\infty) =0.\)
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34B18
zbMATH DE Number: 1398405
zbMATH Keywords: positive solution
zbMATH Keywords: heteroclinic and homoclinic solution
zbMATH Keywords: lower bound for \(c\)
author: Luis Sanchez
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
cites work: An elementary approach to traveling front solutions to a system of N competition-diffusion equations
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: A Note on Degree Theory for Gradient Mappings
cites work: Multidimensional nonlinear diffusion arising in population genetics
cites work: Trajectories joining critical points
cites work: Analytic approximations of travelling wave solutions
cites work: Q5768532
full work available at URL: s0377-0427(99)00254-x
OpenAlex ID: W2162270727
Wikidata QID: Q127990680
description: scientific article; zbMATH DE number 1398405