PreprintCC BY 4.0arXiv
Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range
Abstract
We study non-negative weak solutions to a class of doubly nonlinear nonlocal parabolic equations of the form (uq)t+ℒu=0, where 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 p̃c<p<q+1, where p̃c 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.
Related
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BibTeX
@misc{savchenko2026harnacktype,
title = {{Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range}},
author = {Mariia Savchenko and Igor Skrypnik and Yevgeniia Yevgenieva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29553},
eprint = {2610.29553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29553},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range
AU - Mariia Savchenko
AU - Igor Skrypnik
AU - Yevgeniia Yevgenieva
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29553
UR - https://arxiv.org/abs/2610.29553
AB - We study non-negative weak solutions to a class of doubly nonlinear nonlocal parabolic equations of the form (u^q)_t+ℒu=0, where 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 p̃_c<p<q+1, where p̃_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.
ER -