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Parabolic Frequency for Doubly Nonlinear Equations on Manifolds

Jin Sun, Philipp Sürig

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

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 on weighted complete Riemannian manifolds without any curvature assumption, where denotes the weighted -Laplacian and , . As a consequence, we obtain results on backward uniqueness for and unique continuation at infinity for . 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 .
Paper Structure (4 sections, 13 theorems, 99 equations)

This paper contains 4 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Parabolic frequency on manifolds
  3. More general operators
  4. Liouville-type Theorem for Ancient Solutions

Key Result

Theorem 1

Let $u$ satisfy $\partial_t u = \mathcal{L}_{p,\varphi}u^q$ and assumption E:Assumption. Then In particular, if $\delta\ge 0$, then $N$ is monotone increasing. Moreover, if $\delta=0$, then $\log I(t)$ is convex. If $\delta\neq 0$, then $-\delta^{-1} I(t)^{-\delta/(q+1)}$ is convex.

Theorems & Definitions (27)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Corollary 3
  • Theorem 3
  • Remark 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:static']}
  • ...and 17 more