Parabolic Frequency for Doubly Nonlinear Equations on Manifolds

TL;DR

This paper develops monotonicity formulas for a parabolic frequency function related to doubly nonlinear equations on weighted manifolds, leading to uniqueness, continuation, and Liouville-type results without curvature restrictions.

Contribution

It introduces new monotonicity formulas for parabolic frequencies on manifolds and applies them to establish uniqueness, continuation, and Liouville theorems for doubly nonlinear equations.

Findings

Monotonicity formulas for sign-changing solutions on weighted manifolds.

Backward uniqueness and unique continuation results.

Liouville-type theorems for ancient solutions.

Abstract

We establish monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to a class of doubly nonlinear parabolic equations of the form $\partial_t u = \mathcal{L}_{p,\varphi} u^q$ on weighted complete Riemannian manifolds without any curvature assumption, where $\mathcal{L}_{p,\varphi}$ denotes the weighted $p$-Laplacian and $p>1$, $q>0$. As a consequence, we obtain results on backward uniqueness for $q(p-1)\geq 1$ and unique continuation at infinity for $q(p-1) > 1$. We further consider equations with a controlled nonlinear perturbation term and derive an almost-monotonicity formula for the parabolic frequency. By employing the parabolic frequency, we also establish some Liouville-type results for ancient solutions in the case $q(p-1)\geq 1$.