Existence results for Leibenson's equation on Riemannian manifolds

Abstract

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 $p>1$ and $q>0$ 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 $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.