*preprint*

**Inserted:** 18 jan 2023

**Year:** 2023

**Abstract:**

We investigate local properties of weak solutions to nonlocal and nonlinear
kinetic equations whose prototype is given by
$$
\partial_{t} u +v\cdot\nabla_{x} u +(-\Delta_{v)}^{s}_{p} u = f(u).
$$
We consider equations whose diffusion part is a (possibly degenerate)
integro-differential operator of differentiability order $s \in (0,1)$ and
summability exponent $p\in (1,\infty)$. Amongst other results, we provide an
explicit local boundedness estimate by combining together a suitable Sobolev
embedding theorem and a fractional Caccioppoli-type inequality with tail. For
this, we introduce in the kinetic framework a new definition of nonlocal tail
of a function and of its related tail spaces, also by establishing some useful
estimates for the tail of weak solutions. Armed with the aforementioned results
we give a precise control of the long-range interactions arising from the
nonlocal behaviour of the involved diffusion operator.