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We consider in this paper the maximization problem for the quantity $\int_\Omega u(T, x)dx with respect to $u_0 $, where $u$ is the solution of a given reaction-diffusion equation. This problem is motivated by biological conservation questions. We show the existence of a maximizer and derive optimality conditions through an adjoint problem. We have to face regularity issues since non-smooth initial data could give a better result than smooth ones. We then derive an algorithm enabling to approximate the maximizer and discuss some open problems.

On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data