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arXiv · 2610.29553

Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range

Mariia Savchenko·Igor Skrypnik·Yevgeniia Yevgenieva
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Abstract

We study non-negative weak solutions to a class of doubly nonlinear nonlocal parabolic equations of the form $$ (u^q)_t+\mathcal Lu=0, $$ where $\mathcal L$ is a nonlocal $p$-Laplace-type operator with a symmetric measurable kernel. We consider the fast-diffusion range $1<p<q+1$ and establish Harnack-type estimates in the upper fast-diffusion regime $\widetilde p_c<p<q+1$, where $\widetilde p_c$ is a critical exponent determined by the interplay between the temporal nonlinearity and the nonlocal diffusion. In particular, the estimates apply to the nonlocal parabolic $p$-Laplace equation and the nonlocal porous medium equation.

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Mariia Savchenko, Igor Skrypnik, Yevgeniia Yevgenieva. 2026-10-21. Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range. https://arxiv.org/abs/2610.29553

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