Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917981946118144 |
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| author | Benigno, Serena |
| author_facet | Benigno, Serena |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions Benigno, Serena Analysis of PDEs 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. |
| title | Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.07577 |