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Li-Yau Estimates and Harnack Inequalities for Nonlinear Slow Diffusion Equations on a Smooth Metric Measure Space

Ali Taheri, Vahideh Vahidifar

TL;DR

This work extends Li-Yau and Aronson-Bénilan gradient estimates to positive solutions of nonlinear slow diffusion equations on evolving smooth metric measure spaces with weighted Laplacian $\Delta_\phi$ and fixed exponent $p>1$. The authors derive two main differential Harnack-type bounds under Bakry-Émery curvature and metric-evolution bounds, capturing the interplay between geometry, nonlinearity, and diffusion. These local and global gradient estimates yield parabolic Harnack inequalities and have implications for short- and long-time diffusion dynamics on weighted manifolds, including Liouville-type results and heat kernel bounds. The results generalize previous non-evolving and unweighted cases, providing a unified framework for porous medium type equations on dynamical geometric backgrounds.

Abstract

We present new gradient estimates and Harnack inequalities for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space $(\mathscr M,g,dμ)$ with invariant weighted measure $dμ=e^{-φ} dv_g$ and diffusion operator $Δ_φ=e^φ{\rm div} (e^{-φ} \nabla)$ -- the $φ$-Laplacian. The nonlinear slow diffusion equation, then, for $x \in {\mathscr M}$ and $t>0$, and fixed exponent $p>1$, takes the form \begin{equation*} \partial_t u (x,t) - Δ_φu^p (x,t) = \mathscr N (t,x,u(x,t)). \end{equation*} We assume that the metric tensor $g$ and potential $φ$ are space-time dependent; hence the same is true of the usual metric and potential dependent differential operators and curvature tensors. The estimates are established under natural lower bounds on the Bakry-Émery $m$-Ricci curvature tensor and the time derivative of metric tensor. The curious interplay between geometry, nonlinearity and evolution and their influence on the estimates is at the centre of this investigation. The results here considerably extend and improve earlier results on slow diffusion equations. Several implication, special cases and corollaries are presented and discussed.

Li-Yau Estimates and Harnack Inequalities for Nonlinear Slow Diffusion Equations on a Smooth Metric Measure Space

TL;DR

This work extends Li-Yau and Aronson-Bénilan gradient estimates to positive solutions of nonlinear slow diffusion equations on evolving smooth metric measure spaces with weighted Laplacian and fixed exponent . The authors derive two main differential Harnack-type bounds under Bakry-Émery curvature and metric-evolution bounds, capturing the interplay between geometry, nonlinearity, and diffusion. These local and global gradient estimates yield parabolic Harnack inequalities and have implications for short- and long-time diffusion dynamics on weighted manifolds, including Liouville-type results and heat kernel bounds. The results generalize previous non-evolving and unweighted cases, providing a unified framework for porous medium type equations on dynamical geometric backgrounds.

Abstract

We present new gradient estimates and Harnack inequalities for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space with invariant weighted measure and diffusion operator -- the -Laplacian. The nonlinear slow diffusion equation, then, for and , and fixed exponent , takes the form \begin{equation*} \partial_t u (x,t) - Δ_φu^p (x,t) = \mathscr N (t,x,u(x,t)). \end{equation*} We assume that the metric tensor and potential are space-time dependent; hence the same is true of the usual metric and potential dependent differential operators and curvature tensors. The estimates are established under natural lower bounds on the Bakry-Émery -Ricci curvature tensor and the time derivative of metric tensor. The curious interplay between geometry, nonlinearity and evolution and their influence on the estimates is at the centre of this investigation. The results here considerably extend and improve earlier results on slow diffusion equations. Several implication, special cases and corollaries are presented and discussed.

Paper Structure

This paper contains 7 sections, 18 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Evolution lemmas and the Harnack quantity $F$
  4. Proof of the local estimate in Theorem \ref{['thm5.1']}
  5. Proof of the Harnack inequality in Corollary $\ref{['paraHar']}$
  6. Proof of the local estimate in Theorem \ref{['Thrm-2.5']}
  7. Proof of the Harnack inequality in Corollary $\ref{['corollary-Harnack-global.n']}$

Key Result

Theorem 2.1

Let $(\mathscr M, g, d\mu)$ be a complete smooth metric measure space with $d\mu=e^{-\phi} dv_g$. Suppose that the metric and potential are time dependent, of class $\mathscr{C}^2$ and that for suitable constants $k, \underline k_1, \overline k_1, k_2\ge 0$, $\ell_1, \ell_2 \ge 0$ and $n \le m < \in Here $M = \sup v$ over $Q_{2R,T}$, $c_1$, $c_2 >0$ are the constants in 9.30 and the non-negative q

Theorems & Definitions (28)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Theorem 2.10
  • ...and 18 more