Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

Abraham Solar

doi:10.3934/dcds.2019255

Article Contents

2019, Volume 39, Issue 10: 5799-5823. Doi: 10.3934/dcds.2019255

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Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

Abraham Solar^1,2,

1.
Instituto de Física, Facultad de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
2.
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

Received: August 31, 2018

Revised: March 31, 2019

Published: July 2019

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Abstract

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\mathbb{R}^d}K(y)g(v(t-h,x-y))dy, x \in \mathbb{R}^d,\ t >0;$ where $h>0$ and $d\in\mathbb{Z}_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d = 1$ which yields to the global stability in Sobolev spaces with bounded weights. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semi-wavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.

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Mathematics Subject Classification: Primary: 35K57, 35R10; Secondary: 35B40, 92D25.

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