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

Calc. Var. (2026) 65:197 https://doi.org/10.1007/s00526-026-03371-9 Calculus of Variations Integral Harnack estimates and the rate of extinction of singular fractional diffusion Filippo Maria Cassanello1 · Simone Ciani2 · Antonio Iannizzotto1 Received: 8 October 2025 / Accepted: 26 May 2026 © The Author(s) 2026 Abstract We prove several integral Harnack-type inequalities for local weak solutions of parabolic equations with measurable and bounded coefficients, describing singular s-fractional pLaplacian diffusion. Then we apply such 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. Mathematics Subject Classification 35K67 · 35B65 · 35K92 · 35Q35 1 Introduction and main results 1.1 Heuristics When describing the flow of a non-Newtonian fluid in a simple situation (as in a pipe), the momentum balance law written for a power-like stress tensor can prescribe a dissipation of energy that distinguishes between dilatant fluids, which starting still, stay immobile until a time T ∗ and pseudoplastic fluids, that become immobile after a finite time T ∗ has passed (see [4] Ch.IV Section 7.6). We are here interested in the latter phenomenon (as opposite to the former one), which we rephrase under the more general principle of extinction of a diffusion process after a finite time. As anticipated, this principle is a consequence of the dissipation of energy involved in the evolutive process, that in general can be supplied by a particular Communicated by A. Mondino. B Filippo Maria Cassanello Simone Ciani Antonio Iannizzotto 1 Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy 2 Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, Italy 0123456789().: V,-vol 123 197 Page 2 of 54 F.M. Cassanello et al. source or, as in our case, by the unbalance between the energy of the propagation (parabolic energy terms with power-growth ≈ 2) and the one of the diffusion (elliptic energy terms with a slower power-growth ≈ p < 2). We refer to the classic books [4, 5] for a presentation of energy methods for the study of the localization of the solutions to parabolic equations. When the aim is the description of materials with memory or media that exhibit long-range elastic or plastic deformation, these models account for the fact that stresses or strains at one point in a material can be influenced by other regions over a nonlocal range, rather than just the local neighborhood (see for instance [6, 10, 12, 27, 38, 42] and references therein) and the diffusion is termed nonlocal. It is the precise scope of this work to address the study of the interplay between local and nonlocal effects caused by the phenomenon of extinction and the regularity properties of weak solutions of diffusion processes described, in particular, by the fractional p-Laplacian equation ˆ u t (x, t) = |u(y, t) − u(x, t)| p−2 (u(y, t) − u(x, t)) μ(x, y, t) dy , |x − y| N +sp RN x ∈ , t ∈ [0, T ], (1.1) where p ∈ (1, 2), s ∈ (01), and μ is a measurable bounded function, prescribed by the anisotropy of the medium and reflecting the impossibility to measure the properties described by the model without interfering with themselves (see [23] discussion at 3.1 Ch.I). 1.2 Framing of the Topic and Main Results We consider the following parabolic nonlinear, nonlocal equation u t + L K u = 0, (1.2) in the cylindrical set T =  × [0, T ], with  ⊂ R N open (N  2). The diffusion operator L K is formally defined by ˆ L K u(x, t) = 2 lim |u(x, t) − u(y, t)| p−2 (u(x, t) − u(y, t))K (x, y, t) dy, ε→0+ Bεc (x) where p ∈ (1, 2), s ∈ (0, 1), and K : R N × R N × (0, T ) → R is a measurable function satisfying, for constants 0 < C1  C2 , the following structural properties almost everywhere: (K 1 ) K (x, y, t) = K (y, x, t); (K 2 ) C1  K (x, y, t)|x − y| N + ps  C2 . We remark that when C1 = C2 = 1, the operator L K reduces to the prototype s-fractional p-Laplacian ˆ (−)sp u(x, t) = 2 lim ε→0+ Bεc (x) |u(x, t) − u(y, t)| p−2 (u(x, t) − u(y, t)) dy. |x − y| N + ps For the precise definition of solution adopted we refer to Section 2. For some related results in the elliptic framework, see [13, 26] and the survey paper [28]. Our choice of p ∈ (1, 2) qualifies the diffusion operator as singular, since when u(x, t) = u(y, t) the elliptic term of the diffusion dominates the process. The former quality of the operator has significant 123 Integral Harnack estimates and the rate… Page 3 of 54 197 consequences on the properties of the solutions. Indeed, consider the associated CauchyDirichlet problem: ⎧ ⎪ ⎨u t + L K u = 0 in T (1.3) u=0 in c × (0, T ) ⎪ ⎩ N in R , u(·, 0) = u 0 s, p with  bounded and initial datum u 0 ∈ W0 (), i.e., u 0 ∈ W s, p (R N ) and u 0 = 0 in c (see Section 2 for details). We will prove that (global) weak solutions to (1.3) extinguish within a time T∗ that depends on some L q norm of u 0 (see Theorem 1.5 for the precise statement), while local1 weak solutions satisfy an integral Harnack-type inequality such as, for fixed ρ, t > 0,   1 ˆ ˆ 2− p t γ −1 sup u(x, τ ) d x ≤ inf u(x, τ ) d x + +T, γ > 1, λ 1 0<τ <t B2ρ ρ 0<τ <t Bρ where the second term on the right-hand side takes into account the possible global effects in time, while the term T involves the long-range values of the solution in space, through the quantity ˆ 1  |u(x, τ )| p−1 p−1 ps d x , (1.4) Tail u, x0 , ρ, t1 , t2 = sup ρ N + ps t1 <τ <t2 Bρc (x0 ) |x − x 0 | for fixed 0 < t1 ≤ t2 ≤ T , x0 ∈ , that we refer to as the nonlocal tail of u : R N ×(0, T ) → R (see for instance [24, 25] for the origin of the term, and [35] for an alternative definition of tail and its consequences). Both the property of extinction and this integral Harnack-type inequality are a consequence of the fact that the operator is singular, paralleling the description of the pseudoplastic fluids of Section 1.1. We address the previous integral Harnack inequality, following the terminology of [19], as an L 1 -L 1 Harnack inequality, since it is an Harnack-type estimate for the function t → u(·, t) L 1 (Bρ ) . Another peculiarity of the range 1 < p < 2 is the fact that local weak solutions are not necessarily locally bounded (see [19] Ch. V for a comparison with the p-Laplacian case). A byproduct of our analysis shows with a quantitative estimate that, if p, s satisfy the following relation 2N =: pc < p < 2, N + 2s (1.5) then local weak solutions to (1.2) are locally bounded, provided that the tail terms are adequately controlled (see Remark 3.2 below for the details). Therefore we address the exponent pc as critical, (...truncated)