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

TL;DR

The paper addresses finite extinction time for weak subsolutions of the doubly nonlinear parabolic equation $\rho\partial_t u=\Delta_p u^q$ on complete Riemannian manifolds, under a (weighted) Sobolev inequality with weight $\omega$ and the assumption $D=1-q(p-1)>0$. The authors develop a Caccioppoli-type inequality and introduce a Lyapunov-type functional $\Phi(t)=\int_M u^{\alpha+D}\omega \,dvol$, deriving a differential inequality that leads to finite extinction time via an explicit ODE comparison; the framework encompasses Cartan–Hadamard manifolds (with $\omega\equiv1$) and unweighted cases. They connect to and extend existing results by constructing weighted models that ensure finite extinction for the unweighted Leibenson equation and by applying weighted Sobolev inequalities in Cartan–Hadamard and non-negative Ricci curvature settings. The significance lies in broadening finite-extinction results to diverse geometric contexts through weighted Sobolev inequalities and explicit integrability conditions, providing concrete criteria on exponents and weights and linking to model constructions. These results offer a unified approach to extinction phenomena on manifolds with controlled geometry and contribute to the understanding of nonlinear diffusion in geometric analysis.

Abstract

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}.

Theorems & Definitions (20)

• Theorem 1.1
• Conjecture 1.2
• Definition 1
• Lemma 2.1: Caccioppoli type inequality
• Definition 2
• Definition 3
• Theorem 3.1
• Remark 1
• Lemma 4.1
• Example 1