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Integral Harnack estimates and the rate of extinction of singular fractional diffusion

Filippo M. Cassanello, Simone Ciani, Antonio Iannizzotto

TL;DR

The paper investigates singular fractional diffusion driven by a nonlocal parabolic operator with a bounded, symmetric kernel, proving integral Harnack-type inequalities and finite extinction phenomena for weak solutions. The core methodology combines time mollification with energy estimates and a tailored De Giorgi-type iteration, yielding sharp $L^r$-$L^{\infty}$ and $L^1$-$L^1$/$L^1$-$L^{\infty}$ estimates that incorporate nonlocal tail terms. Extinction times are established for the Cauchy-Dirichlet problem, with explicit formulas depending on the critical exponent $p_c$ and the initial data, and the decay rates toward extinction are quantified in terms of time-to-extinction and spatial scales. These results extend the Harnack framework to singular fractional diffusion with general kernels, providing tools for regularity, traces, and long-range behavior in nonlocal parabolic problems with potential applications to materials with memory and nonlocal diffusion phenomena.

Abstract

We prove several integral Harnack-type inequalities for local weak solutions of parabolic equations with measurable and bounded coefficients, describing singular s-fractional p-Laplacian diffusion. Then we apply the aforementioned estimates to evaluate the decay rate of the local mass and supremum of the solutions as they approach a possible extinction time. Yet we show consistency of our general decay estimates by studying the extinction phenomenon for weak solutions of the Cauchy-Dirichlet problem, by means of an approximation procedure that carefully avoids the use of an integrable time derivative.

Integral Harnack estimates and the rate of extinction of singular fractional diffusion

TL;DR

The paper investigates singular fractional diffusion driven by a nonlocal parabolic operator with a bounded, symmetric kernel, proving integral Harnack-type inequalities and finite extinction phenomena for weak solutions. The core methodology combines time mollification with energy estimates and a tailored De Giorgi-type iteration, yielding sharp - and -/- estimates that incorporate nonlocal tail terms. Extinction times are established for the Cauchy-Dirichlet problem, with explicit formulas depending on the critical exponent and the initial data, and the decay rates toward extinction are quantified in terms of time-to-extinction and spatial scales. These results extend the Harnack framework to singular fractional diffusion with general kernels, providing tools for regularity, traces, and long-range behavior in nonlocal parabolic problems with potential applications to materials with memory and nonlocal diffusion phenomena.

Abstract

We prove several integral Harnack-type inequalities for local weak solutions of parabolic equations with measurable and bounded coefficients, describing singular s-fractional p-Laplacian diffusion. Then we apply the aforementioned estimates to evaluate the decay rate of the local mass and supremum of the solutions as they approach a possible extinction time. Yet we show consistency of our general decay estimates by studying the extinction phenomenon for weak solutions of the Cauchy-Dirichlet problem, by means of an approximation procedure that carefully avoids the use of an integrable time derivative.
Paper Structure (15 sections, 20 theorems, 264 equations, 1 figure)

This paper contains 15 sections, 20 theorems, 264 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Heuristics
  3. Framing of the Topic and Main Results
  4. Main Results
  5. Novelty and Significance
  6. The Method
  7. Future Applications
  8. Structure of the paper
  9. Notation
  10. Preliminaries
  11. $L^r$-$L^\infty$ estimate
  12. $L^1$-$L^1$ and $L^1$-$L^{\infty}$ estimates
  13. Backward $L^r-L^r$ estimate
  14. Extinction time and decay estimates
  15. Time mollifications

Key Result

Theorem 1.1

($L^r$-$L^\infty$ estimate) Let $r\geqslant 1$ be such that $\lambda_r>0$, with $1\leq r<2$ when $p>p_c$. If $u$ is a locally bounded solution of eq, nonnegative in $B_{4\rho}(x_0)\times(0,t)\subset\Omega_T$, then there exists $\gamma>0$ depending on the data and $r$, s.t.

Figures (1)

  • Figure 1: Illustration of the extinction decay for a local weak solution to \ref{['cdp']}.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 23 more