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Existence and stability of bistable wavefronts in a nonlocal delayed reaction–diffusion epidemic system

Part of: Miscellaneous topics - Partial differential equations Parabolic equations and systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  24 March 2020

KUN LI  and
XIONG LI
KUN LI
Affiliation:
School of Mathematics and Computational Science, Hunan First Normal University, Changsha, Hunan410205, People’s Republic of China email: kli@mail.bnu.edu.cn
XIONG LI
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing100875, People’s Republic of China email: xli@bnu.edu.cn
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Abstract

In this paper, we consider the monotone travelling wave solutions of a reaction–diffusion epidemic system with nonlocal delays. We obtain the existence of monotone travelling wave solutions by applying abstract existence results. By transforming the nonlocal delayed system to a non-delayed system and choosing suitable small positive constants to define a pair of new upper and lower solutions, we use the contraction technique to prove the asymptotic stability (up to translation) of monotone travelling waves. Furthermore, the uniqueness and Lyapunov stability of monotone travelling wave solutions will be established with the help of the upper and lower solution method and the exponential asymptotic stability.

Keywords

Travelling wave frontnonlocal delaycontraction techniqueupper and lower solutions

MSC classification

Primary: 35R20: Partial operator-differential equations (i.e., PDE on finite-dimensional spaces for abstract space valued functions)
Secondary: 35K57: Reaction-diffusion equations 34K30: Equations in abstract spaces
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 146 - 176
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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