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

We consider on Riemannian manifolds the non-linear evolution equation $$ρ\partial {t}u=Δ{p}u^{q}.$$ Assuming that the manifold satisfies a \textit{(weighted) Sobolev inequality} and under certain assumptions on $p, q$ and function $ρ$, 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}.

💡 Research Summary

The paper studies the doubly nonlinear parabolic equation
  ρ ∂ₜ u = Δₚ uᵠ  (p > 1, q > 0)
on a complete Riemannian manifold (M,g). Here Δₚ denotes the p‑Laplacian, ρ > 0 is a spatial weight, and u ≥ 0 is the unknown. The authors focus on the regime D := 1 − q(p − 1) > 0, which guarantees that the nonlinearity is “sub‑critical” in the sense that the equation admits finite‑time extinction for suitable solutions.

Weak subsolutions.
A function u is called a weak subsolution if it belongs to C(