KPP reaction-diffusion systems with loss inside a cylinder: convergence toward the problem with Robin boundary conditions

TL;DR

This paper studies reaction-diffusion systems with heat loss inside a cylindrical domain, analyzing how solutions and minimal speeds converge to a boundary heat loss model with Robin boundary conditions, especially in two dimensions.

Contribution

It establishes the convergence of minimal speeds and traveling front solutions from interior heat loss models to boundary heat loss models with Robin conditions.

Findings

Minimal speeds converge to the boundary heat loss problem's minimal speed.

Traveling front solutions converge under certain assumptions, valid in 2D.

Results connect interior and boundary heat loss models in reaction-diffusion systems.

Abstract

We consider in this paper a reaction-diffusion system under a KPP hypothesis in a cylindrical domain in the presence of a shear flow. Such systems arise in predator-prey models as well as in combustion models with heat losses. Similarly to the single equation case, the existence of a minimal speed c* and of traveling front solutions for every speed c > c* has been shown both in the cases of heat losses distributed inside the domain or on the boundary. Here, we deal with the accordance between the two models by choosing heat losses inside the domain which tend to a Dirac mass located on the boundary. First, using the characterizations of the corresponding minimal speeds, we will see that they converge to the minimal speed of the limiting problem. Then, we will take interest in the convergence of the traveling front solutions of our reaction-diffusion systems. We will show the convergence under some assumptions on those solutions, which in particular can be satisfied in dimension 2.