Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition

Tarik Mohammed Touaoula

doi:10.3934/cpaa.2020108

Article Contents

2020, Volume 19, Issue 5: 2473-2490. Doi: 10.3934/cpaa.2020108

This issue Previous Article Uniform a priori estimates for elliptic problems with impedance boundary conditions Next Article Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy

Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition

Tarik Mohammed Touaoula

Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, ALGERIA

Received: October 2018

Revised: October 2019

Published: March 2020

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Abstract

In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous Neumann boundary condition. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument based on sub and super-solutions combined with the fluctuation method. We also give a condition under which the exponential stability of the positive steady state is reached. As particular examples, we apply our results to the diffusive Nicholson blowfly equation and the diffusive Mackey-Glass equation with distributed delay. We obtain some new results on exponential stability of the positive steady state for these models.

Keywords:

Reaction-diffusion equation; distributed delay,
sub and super-solution,
global attractivity; exponential stability.

Mathematics Subject Classification: Primary: 35R10, 35B40, 35Q92.

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