Interior Regularity of Mixed Local-Nonlocal Parabolic Semilinear Equations
Resumen:
In this paper we prove the existence, uniqueness and regularity of classical solutions \break of a semilinear parabolic equation with a mixed local and nonlocal diffusion operator \break $\mathcal{L} = -(-\Delta)^s + \Delta$ and Dirichlet boundary conditions. Here, $(-\Delta)^s$ is the integral fractional laplacian and $\Delta$ is the classic local laplacian. We then study the interior regularity of said solutions and conclude that they are H\"older continuous in both space and time, and they are $C^{2,\alpha}_{loc}$ in space for all positive times.
| 2026 | |
| Agencia Nacional de Investigación e Innovación | |
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Regularity Mixed parabolic problem Nonlocal diffusion Fractional laplacian Mixed operator Interior regularity Hölder continuity Ciencias Naturales y Exactas Matemáticas |
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| Inglés | |
| Agencia Nacional de Investigación e Innovación | |
| REDI | |
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https://hdl.handle.net/20.500.12381/5714
https://doi.org/10.48550/arXiv.2610.06537 |
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| Acceso abierto | |
| Reconocimiento 4.0 Internacional. (CC BY) |
| Sumario: | In this paper we prove the existence, uniqueness and regularity of classical solutions \break of a semilinear parabolic equation with a mixed local and nonlocal diffusion operator \break $\mathcal{L} = -(-\Delta)^s + \Delta$ and Dirichlet boundary conditions. Here, $(-\Delta)^s$ is the integral fractional laplacian and $\Delta$ is the classic local laplacian. We then study the interior regularity of said solutions and conclude that they are H\"older continuous in both space and time, and they are $C^{2,\alpha}_{loc}$ in space for all positive times. |
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