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 ${\partial }_{t}u={\mathcal{L}}_{p,\ddot{I}†}{u}^q$ on weighted complete Riemannian manifolds without any curvature assumption, where ${\mathcal{L}}_{p,\ddot{I}†}$ denotes the weighted $p$-Laplacian and $p>1$, $q>0$. As a consequence, we obtain results on backward uniqueness for $q(p-1)\ge 1$ 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 $q(p-1)\ge 1$.

16 pages. Comments are welcome!