A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations
We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffu...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-11-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/11/1059 |
| _version_ | 1827640565661958144 |
|---|---|
| author | Tucker Hartland Ravi Shankar |
| author_facet | Tucker Hartland Ravi Shankar |
| author_sort | Tucker Hartland |
| collection | DOAJ |
| description | We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp. |
| first_indexed | 2024-03-09T17:01:28Z |
| format | Article |
| id | doaj.art-46bd3234c59e40fb98fbb85ba6c95d98 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T17:01:28Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-46bd3234c59e40fb98fbb85ba6c95d982023-11-24T14:29:03ZengMDPI AGAxioms2075-16802023-11-011211105910.3390/axioms12111059A Strong Maximum Principle for Nonlinear Nonlocal Diffusion EquationsTucker Hartland0Ravi Shankar1Department of Applied Mathematics, University of California, Merced, CA 95343, USADepartment of Mathematics, Princeton University, Princeton, NJ 08544, USAWe consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp.https://www.mdpi.com/2075-1680/12/11/1059nonlocal diffusionnonlinear diffusionintegro-differential equationsmaximum principlestrong maximum principledegenerate |
| spellingShingle | Tucker Hartland Ravi Shankar A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations Axioms nonlocal diffusion nonlinear diffusion integro-differential equations maximum principle strong maximum principle degenerate |
| title | A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations |
| title_full | A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations |
| title_fullStr | A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations |
| title_full_unstemmed | A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations |
| title_short | A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations |
| title_sort | strong maximum principle for nonlinear nonlocal diffusion equations |
| topic | nonlocal diffusion nonlinear diffusion integro-differential equations maximum principle strong maximum principle degenerate |
| url | https://www.mdpi.com/2075-1680/12/11/1059 |
| work_keys_str_mv | AT tuckerhartland astrongmaximumprinciplefornonlinearnonlocaldiffusionequations AT ravishankar astrongmaximumprinciplefornonlinearnonlocaldiffusionequations AT tuckerhartland strongmaximumprinciplefornonlinearnonlocaldiffusionequations AT ravishankar strongmaximumprinciplefornonlinearnonlocaldiffusionequations |