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Gradient estimates for the weighted porous medium equation on graphs

Shoudong Man

TL;DR

This work extends Li–Yau type gradient estimates to nonlinear, variable-exponent diffusion on graphs by establishing discrete chain rules for the fractional and standard Laplacians and deriving a Li–Yau–type bound for the gradient functional $F(x,t)$. The main contributions include a gradient estimate for the fractional porous medium equation with variable exponent, a Harnack inequality, and a suite of applications: two-sided heat kernel bounds, volume growth estimates, and a Buser-type inequality relating spectral data to isoperimetric properties. The framework unifies nonlinear fractional diffusion with graph geometry and recovers classical linear results in the appropriate limits. Overall, the results significantly generalize prior linear theory on graphs to nonlinear, nonlocal diffusion, with concrete consequences for heat propagation, geometry, and spectral theory on graphs.

Abstract

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$Δu^{m}=δ(x)u_{t}+ψu^{m}$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].

Gradient estimates for the weighted porous medium equation on graphs

TL;DR

This work extends Li–Yau type gradient estimates to nonlinear, variable-exponent diffusion on graphs by establishing discrete chain rules for the fractional and standard Laplacians and deriving a Li–Yau–type bound for the gradient functional . The main contributions include a gradient estimate for the fractional porous medium equation with variable exponent, a Harnack inequality, and a suite of applications: two-sided heat kernel bounds, volume growth estimates, and a Buser-type inequality relating spectral data to isoperimetric properties. The framework unifies nonlinear fractional diffusion with graph geometry and recovers classical linear results in the appropriate limits. Overall, the results significantly generalize prior linear theory on graphs to nonlinear, nonlocal diffusion, with concrete consequences for heat propagation, geometry, and spectral theory on graphs.

Abstract

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation on graphs for , which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].

Paper Structure

This paper contains 10 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Gradient Estimates
  4. Discrete Chain Rules
  5. Gradient Estimates
  6. Harnack Inequalities
  7. Applications
  8. Heat Kernel Estimates
  9. Volume Growth
  10. Buser-Type Inequality

Key Result

Lemma 3.1

Let $G = (V,E,\vartheta,w)$ be a connected, locally finite, and stochastically complete graph equipped with the fractional Laplace operator $\Delta^{\epsilon}$. For any positive function $u: V \to \mathbb{R}^+$ and any variable exponent $m: V \times [0,\infty) \to \mathbb{R}^+$, the following identi

Theorems & Definitions (21)

  • Lemma 3.1: Discrete Chain Rule for the Fractional Laplacian
  • proof
  • Lemma 3.2: Discrete Chain Rule for the Standard Laplacian
  • Remark 3.3
  • Theorem 3.4: Gradient Estimate for the fractional Laplacian in the Variable Exponents case
  • proof
  • Remark 3.5
  • Theorem 3.6: Gradient Estimate for the Standard Laplacian with Variable Exponents
  • proof
  • Lemma 4.1: Extremal Lemma
  • ...and 11 more