The optimal initial datum for a class of reaction-advection-diffusion equations

TL;DR

This paper identifies the optimal initial distribution for a reaction-diffusion system with advection to maximize total mass at a fixed time, showing advection can enhance population growth.

Contribution

It proves existence and uniqueness of the optimal initial data and compares total mass with and without advection, highlighting advection's positive effect.

Findings

Optimal initial data exists and is unique.

Advection increases total mass at time T.

Presence of large advection enhances population growth.

Abstract

We consider a reaction-diffusion model with a drift term in a bounded domain. Given a time $T,$ we prove the existence and uniqueness of an initial datum that maximizes the total mass $\textstyle{\int_\Omega u(T,x)\mathrm{d}x}$ in the presence of an advection term. In a population dynamics context, this optimal initial datum can be understood as the best distribution of the initial population that leads to a maximal the total population at a prefixed time $T.$ We also compare the total masses at a time $T$ in two cases: depending on whether an advection term is present in the medium or not. We prove that the presence of a large enough advection enhances the total mass.