Traveling fronts in a reaction-diffusion equation with a memory term

TL;DR

This paper investigates the existence of traveling fronts in reaction-diffusion equations with memory terms, showing how such terms can arise from homogenization and proving the existence of fronts using comparison principles and reformulation techniques.

Contribution

It introduces a novel approach to analyze reaction-diffusion equations with memory by transforming memory effects into nonlocal spatial terms and establishing conditions for traveling front existence.

Findings

Existence of traveling fronts in memory-affected reaction-diffusion systems.

Memory terms can be derived from homogenization of periodic systems.

Numerical simulations support theoretical results.

Abstract

Based on a recent work on traveling waves in spatially nonlocal reaction-diffusion equations, we investigate the existence of traveling fronts in reaction-diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction-diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising travelingfront, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.