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On the maximization problem for solutions of reaction-diffusion equations with respect to their initial data. (English) Zbl 07372601
Summary: We consider in this paper the maximization problem for the quantity $$_{\int\Omega}u(t,x)\mathrm{d}x$$ with respect to $$u_0=:u(0,\cdot)$$, 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.
##### MSC:
 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35Q93 PDEs in connection with control and optimization 92D25 Population dynamics (general) 92D30 Epidemiology
##### Keywords:
reaction-diffusion; control; conservation biology; optimization
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