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arXiv · 2504.07577

Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions

Serena Benigno
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Abstract

In this paper we study a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on the boundary of the domain. First, we prove that the parabolic problem has a unique positive, bounded solution. Then, we show that this solution converges as t tends to infinity to the unique nonnegative solution of the elliptic associated problem. The existence of the unique positive solution to this problem depends on a principal eigenvalue of a suitable linearized problem with a sign-changing weights. Next, we study the minimization of such eigenvalue with respect to the sign-changing weight, showing that there exists an optimal bang-bang weight, namely a piece-wise constant weight that takes only two values. Finally, we completely solve the problem in dimension one.

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Serena Benigno. 2025-04-10. Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions. https://arxiv.org/abs/2504.07577

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