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

TL;DR

This paper proves that weak subsolutions of a weighted nonlinear evolution equation on Riemannian manifolds, including Cartan-Hadamard manifolds, become extinct in finite time under certain Sobolev inequality conditions.

Contribution

It establishes finite extinction time for subsolutions of a weighted Leibenson equation on Riemannian manifolds, extending results to Cartan-Hadamard manifolds.

Findings

Weak subsolutions have finite extinction time under Sobolev inequality conditions.

Main results apply to Cartan-Hadamard manifolds.

Conditions on p, q, and ρ are crucial for extinction.

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}.