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

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.

💡 Research Summary

The paper investigates a class of nonlinear, nonlocal parabolic equations driven by the singular fractional p‑Laplacian, namely
 u_t + L_K u = 0 in Ω_T = Ω × (0,T),
with 1 < p < 2 and s ∈ (0,1). The diffusion operator L_K is defined through a symmetric measurable kernel K(x,y,t) satisfying the two‑sided bound
 C₁|x−y|^{−N−ps} ≤ K(x,y,t) ≤ C₂|x−y|^{−N−ps}.
Because p < 2 the operator is “singular”: the diffusion term dominates whenever the solution values coincide, a feature that mimics the behavior of pseudoplastic fluids.

The authors’ main contributions are a suite of integral Harnack‑type inequalities and quantitative extinction results, all derived without assuming the existence of an integrable time derivative. The key results can be grouped as follows:

1. Local boundedness and L^r–L^∞ estimate (Theorem 1.1).
   For any r ≥ 1 with λ_r := N(p−2)+rps > 0, a locally bounded weak solution that is non‑negative in a cylinder B_{4ρ}(x₀)×(0,t) satisfies
    sup_{B_{ρ/2}×(t/2,t)} u ≤ γ