Existence results for Leibenson's equation on Riemannian manifolds

We consider on an arbitrary Riemannian manifold $M$ the $\mathit{\text{Leibenson}}\mathit{\text{equation}}$ ${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_p{u}^q$, that is also known as a $\mathit{\text{doubly}}\mathit{\text{nonlinear}}\mathit{\text{evolution}}\mathit{\text{equation}}$. 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 $u_0\in L^1(M)\cap L^{\mathrm{\infty }}(M)$.