Existence results for Leibenson's equation on Riemannian manifolds

TL;DR

This paper proves existence and uniqueness of weak solutions for Leibenson's equation, a doubly nonlinear PDE, on arbitrary Riemannian manifolds under certain conditions.

Contribution

It establishes the existence and uniqueness of solutions for Leibenson's equation on Riemannian manifolds when p>1, q>0, and pq≥1.

Findings

Unique weak solutions exist for initial data in L¹∩L∞.

Solutions are valid for any Riemannian manifold under the specified conditions.

The results extend PDE theory to nonlinear equations on curved spaces.

Abstract

We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=\Delta _{p}u^{q}$, 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=\Delta _{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 $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.