Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds

Abstract

We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=\Delta _{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)<1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)<1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.