Issue |
Math. Model. Nat. Phenom.
Volume 15, 2020
|
|
---|---|---|
Article Number | 71 | |
Number of page(s) | 22 | |
DOI | https://doi.org/10.1051/mmnp/2020030 | |
Published online | 03 December 2020 |
On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data
1
Sorbonne Université, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,
75005
Paris, France.
2
Université Paris Nord, Institut Galilée, UMR 7539, LAGA,
93430
Villetaneuse, France.
* Corresponding author: gregoire.nadin@upmc.fr
Received:
5
July
2019
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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