Minimal travelling wave speed and explicit solutions in monostable reaction-diffusion equations

Elaine Crooks,Michael Grinfeld

doi:10.14232/ejqtde.2020.1.79

Abstract

We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.

Highlights

To Jeff in appreciation and gratitude. In this short paper we investigate the somewhat puzzling connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a monostable reaction-diffusion equation
For parameter-dependent problems with a parameter-dependent family of explicit solutions, there are many cases where there is a switching between the minimal speed being given by this explicit solution for some parameters, while for others it is given by the so-called linear speed, defined as the minimal value for which the problem linearised about the unstable steady state has a suitable eigenvalue
Lemma 3.1 clearly still implies that cmin(β) = cnl(β) > cl(β), so in particular nonlinear selection holds, if A(β) < 2B(β), and linear selection holds, with cmin(β) = cnl(β) = cl(β) if A(β) = 2B(β), but whether it is possible to have again nonlinear selection for some β with A(β) > 2B(β), either with the minimal speed corresponding to the explicit solution or another value, is not obvious

Summary

Introduction

In this short paper we investigate the somewhat puzzling connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a monostable reaction-diffusion equation. We introduce scalar monostable reaction-diffusion equations, define what we mean by a minimal speed, and discuss the linear (pulled) and the non-linear (pushed) regimes. The basis of analysis of monotone fronts in the scalar monostable case (1.2) is the following construction: As U(z) is a monotone solution, its derivative is a well-defined function of U.

Exact solvability

Minimality exchange

Conclusions