Optimization of the L1 norm of the solution of a Fisher-KPP equations in the small diffusivity regime

TL;DR

This paper studies how to maximize the L1 norm of solutions to a Fisher-KPP equation as diffusivity approaches zero, revealing BV properties, limit behaviors, and periodicity of maximizers.

Contribution

It characterizes the limit of maximizers' BV norms and constructs quasi-maximizers with specific periodicity and norm behavior as diffusivity tends to zero.

Findings

Maximizers are always BV functions.

The BV norm of maximizers behaves like the inverse square root of diffusivity.

Maximizers exhibit periodicity along subsequences of diffusivities.

Abstract

We investigate in the present paper the maximization problem for the L1 norm of the unique positive solution of an heterogeneous Fisher-KPP equation with respect to the growth rate. It is already known that the BV norms of maximizers of this functional blow up when the diffusivity tends to zero. Here, we first show that the maximizers are always BV. Next, we completely characterize the limit of the maximas of this functional as the diffusivity tends to zero, and we show that one can construct a quasi-maximizer which is periodic, in a sense, and with a BV norm behaving like the inverse of the square root of the diffusivity. Lastly, we prove that along a subsequence of diffusivities, any maximizer is periodic, in a sense.