Parabolic Frequency for Doubly Nonlinear Equations on Manifolds

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 $q(p-1)\geq 1$.