Harnack-type inequalities for nonlocal doubly nonlinear parabolic equations

We establish intrinsic Harnack inequalities for nonlocal doubly nonlinear parabolic equations in the slow-diffusion range: \begin{equation*} (u^q)t+ \operatorname{P.V.} \int\limits{$\mathbb{R}$^N}\frac{|u(x, t)-u(y, t)|^{p-2}(u(x, t)-u(y, t))}{|x-y|^{N+sp}},dy=0,\quad s\in(0,1),\quad 0<q < p-1. \end{equation*}