A variational approach to reaction-diffusion equations with forced speed in dimension 1

Juliette Bouhours; Grégroie Nadin

doi:10.3934/dcds.2015.35.1843

Article Contents

2015, Volume 35, Issue 5: 1843-1872. Doi: 10.3934/dcds.2015.35.1843

This issue Previous Article Lipschitz perturbations of expansive systems Next Article Stability of singular limit cycles for Abel equations

A variational approach to reaction-diffusion equations with forced speed in dimension 1

Juliette Bouhours^1, and
Grégroie Nadin^1,

1.
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received: October 2013

Revised: October 2014

Published: May 2017

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Abstract

We investigate in this paper a scalar reaction diffusion equation with a nonlinear reaction term depending on $x-ct$. Here, $c$ is a prescribed parameter modelling the speed of climate change and we wonder whether a population will survive or not, that is, we want to determine the large-time behaviour of the associated solution.
This problem has been solved recently when the nonlinearity is of KPP type. We consider in the present paper general reaction terms, that are only assumed to be negative at infinity. Using a variational approach, we construct two thresholds $0<\underline{c}\leq \overline{c} <\infty$ determining the existence and the non-existence of travelling waves. Numerics support the conjecture $\underline{c}=\overline{c}$. We then prove that any solution of the initial-value problem converges at large times, either to $0$ or to a travelling wave. In the case of bistable nonlinearities, where the steady state $0$ is assumed to be stable, our results lead to constrasting phenomena with respect to the KPP framework. Lastly, we illustrate our results and discuss several open questions through numerics.

Keywords:

Mathematics Subject Classification: 35B40, 35C07, 35J20, 35K10, 35K57, 92D52.

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