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

Grégoire Nadin (1) , Ana Isis Toledo Marrero (2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 MATCOM - Matematica y Computacion

We consider in this paper the maximization problem for the quantity ∫ Ωu(t, x)dx with respect to u0 =: 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.

Mots clés: Reaction-diffusion; control; conservation biology; optimization

Domaines: Mathématiques [math]

Grégoire Nadin, Ana Isis Toledo Marrero. On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data. Mathematical Modelling of Natural Phenomena, 2020, 15, pp.71. ⟨10.1051/mmnp/2020030⟩. ⟨hal-03039668v2⟩