Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions
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.
Keywords
Cite
@article{arxiv.2504.07577,
title = {Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions},
author = {Serena Benigno},
journal= {arXiv preprint arXiv:2504.07577},
year = {2025}
}