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Smoothing effects and extinction in finite time for fractional fast diffusions on Riemannian manifolds

Elvise Berchio, Matteo Bonforte, Gabriele Grillo

TL;DR

This work extends fractional fast diffusion to noncompact Riemannian manifolds with curvature and functional-analytic structure. By combining semigroup theory for mild solutions with a fractional Green-function framework, the authors establish existence of Weak Dual Solutions, precise $L^p$-to-$L^\infty$ smoothing (including weighted versions), and infinite-speed propagation, along with sharp finite-time extinction rates. Central to the analysis are Green-function estimates, Kato and Stroock–Varopoulos-type inequalities, and Nash-type fractional inequalities adapted to the manifold setting. The results broaden the Euclidean theory to curved spaces, revealing how geometric inequalities and Green-function behavior govern nonlocal, singular diffusion on manifolds and enabling quantitative extinction and propagation analyses.

Abstract

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power that makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual $L^1$ space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted $L^p$ spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates.

Smoothing effects and extinction in finite time for fractional fast diffusions on Riemannian manifolds

TL;DR

This work extends fractional fast diffusion to noncompact Riemannian manifolds with curvature and functional-analytic structure. By combining semigroup theory for mild solutions with a fractional Green-function framework, the authors establish existence of Weak Dual Solutions, precise -to- smoothing (including weighted versions), and infinite-speed propagation, along with sharp finite-time extinction rates. Central to the analysis are Green-function estimates, Kato and Stroock–Varopoulos-type inequalities, and Nash-type fractional inequalities adapted to the manifold setting. The results broaden the Euclidean theory to curved spaces, revealing how geometric inequalities and Green-function behavior govern nonlocal, singular diffusion on manifolds and enabling quantitative extinction and propagation analyses.

Abstract

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power that makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates.
Paper Structure (16 sections, 15 theorems, 87 equations)

This paper contains 16 sections, 15 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Geometric assumptions and functional setting
  3. Fractional Laplacian, fractional potentials and related functional inequalities
  4. Notation
  5. Main results
  6. Definition of Weak Dual Solutions (WDS), and their existence
  7. Statement of the main results concerning smoothing effects
  8. Lower bound and infinite speed of propagation
  9. Technical results
  10. Fractional Green function estimates
  11. Key inequalities
  12. Proof of Theorem \ref{['Thm.Smoothing']}
  13. Proof of Theorem \ref{['ext']}
  14. Proof of Theorem \ref{['Thm.SmoothingPhi']}
  15. Proof of Theorem \ref{['lowerth']}
  16. ...and 1 more sections

Key Result

Proposition 1.4

(see Caselli) Let $M$ be a complete, stochastically complete Riemannian manifold. For all $v\in C_c^\infty(M)$, for all $x\in M$, one has: with where $c_s=1/\Gamma(-s)$. If $s<1/2$ the integral is absolutely convergent, hence the principle value is not required.

Theorems & Definitions (22)

  • Proposition 1.4
  • Remark 1.1
  • Proposition 1.5
  • Corollary 1.6
  • Definition 2.1
  • Theorem 2.1: Existence of a WDS for data in $L^1_{\mathbb{G}_{M}^s}$
  • Corollary 2.2: Uniqueness of limit WDS
  • Theorem 2.3: $L^p-L^\infty$ smoothing
  • Theorem 2.4: Extinction Time
  • Proposition 2.5: Sharp $L^{1+m}$ extinction rate
  • ...and 12 more