  In this paper we investigate the nonexistence of nonnegative solutions of
parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq
(K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq
1,\\ &u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad
(P^{\pm}) $$ where $u_0\in L^1_{loc}({\mathbb R}^N)$, $L_{\mathcal{A}}$ denotes
a weakly $m$-coercive operator, which includes as prototype the $m$-Laplacian
or the generalized mean curvature operator, $p,\,q>0$, while $K\ast u^p$ stands
for the standard convolution operator between a weight $K>0$ satisfying
suitable conditions at infinity and $u^p$. For problem $(P^-)$ we obtain a
Fujita type exponent while for $(P^+)$ we show that no such critical exponent
exists. Our approach relies on nonlinear capacity estimates adapted to the
nonlocal setting of our problems. No comparison results or maximum principles
are required.
