Existence results for Leibenson's equation on Riemannian manifolds
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910134054158336 |
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| author | Sürig, Philipp |
| author_facet | Sürig, Philipp |
| contents | We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=Δ_{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1, q>0$ and $pq\geq 1$ then the Cauchy-problem \begin{equation*} \left\{\begin{array}{ll}\partial _{t}u=Δ_{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M,\end{array}\right.\end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20640 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence results for Leibenson's equation on Riemannian manifolds Sürig, Philipp Analysis of PDEs Differential Geometry 35K55, 58J35, 53C21, 35B05 We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=Δ_{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1, q>0$ and $pq\geq 1$ then the Cauchy-problem \begin{equation*} \left\{\begin{array}{ll}\partial _{t}u=Δ_{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M,\end{array}\right.\end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$. |
| title | Existence results for Leibenson's equation on Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry 35K55, 58J35, 53C21, 35B05 |
| url | https://arxiv.org/abs/2601.20640 |