The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913758409916416 |
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| author | Grillo, Gabriele Monticelli, Dario D. Punzo, Fabio |
| author_facet | Grillo, Gabriele Monticelli, Dario D. Punzo, Fabio |
| contents | We consider the porous medium equation (PME) on complete noncompact manifolds $M$ of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space $X$ of functions, strictly larger than $L^1$, in which the Green function on $M$ appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to $X$. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to $X$ as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach Grillo, Gabriele Monticelli, Dario D. Punzo, Fabio Analysis of PDEs Differential Geometry Functional Analysis We consider the porous medium equation (PME) on complete noncompact manifolds $M$ of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space $X$ of functions, strictly larger than $L^1$, in which the Green function on $M$ appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to $X$. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to $X$ as well. |
| title | The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach |
| topic | Analysis of PDEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2402.18706 |