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

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.