Math. Model. Nat. Phenom.
Volume 15, 2020
|Number of page(s)||22|
|Published online||03 December 2020|
On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data
Sorbonne Université, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,
2 Université Paris Nord, Institut Galilée, UMR 7539, LAGA, 93430 Villetaneuse, France.
* Corresponding author: email@example.com
Accepted: 27 July 2020
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.
Mathematics Subject Classification: 35B30 / 35B45 / 35B65 / 35K15 / 35K57 / 35Q80 / 35Q92 / 35Q93 / 92D25 / 92D30
Key words: Reaction-diffusion / control / conservation biology / optimization
© The authors. Published by EDP Sciences, 2020
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