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

We consider on Riemannian manifolds the non-linear evolution equation $$\rho {\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_p{u}^q\mathrm{.}$$ Assuming that the manifold satisfies a $\mathit{\text{(weighted)}}\mathit{\text{Sobolev}}\mathit{\text{inequality}}$ 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 $\mathit{\text{Cartan-Hadamard}}\mathit{\text{manifold}}$.