Global existence for a Leibenson type equation with reaction on Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2025
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| author | Meglioli, Giulia Oliva, Francescantonio Petitta, Francesco |
| author_facet | Meglioli, Giulia Oliva, Francescantonio Petitta, Francesco |
| contents | We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = Δ_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global existence for a Leibenson type equation with reaction on Riemannian manifolds Meglioli, Giulia Oliva, Francescantonio Petitta, Francesco Analysis of PDEs We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = Δ_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure. |
| title | Global existence for a Leibenson type equation with reaction on Riemannian manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.08304 |