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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
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