Finite extinction time for subsolutions of the weighted Leibenson equation on Riemannian manifolds

Abstract

We consider on Riemannian manifolds the non-linear evolution equation $$\rho \partial _{t}u=\Delta _{p}u^{q}.$$ Assuming that the manifold satisfies a \textit{(weighted) Sobolev inequality} and under certain assumptions on $p, q$ and function $\rho$, we prove that weak subsolutions to this equation have a finite extinction time. In particular, our main result holds in the case of a \textit{Cartan-Hadamard manifold}.