  We study the behavior as $t\to 0^+$ of nonnegative functions
\begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap
L^\lambda (\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying
the parabolic Choquard-Pekar type inequalities \begin{equation}\label{0.2}
  0\leq u_t-\Delta u\leq(\Phi^{\alpha/n}*u^\lambda )u^\sigma \quad \text{ in
}B_1 (0)\times (0,1) \end{equation} where $\alpha\in(0,n+2)$, $\lambda>0$, and
$\sigma\geq0$ are constants, $\Phi$ is the heat kernel, and $*$ is the
convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal
conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions
$u$ satisfy pointwise bounds in compact subsets of $B_1(0)$ as $t\to 0^+$. We
obtain similar results for nonnegative solutions when $\Phi^{\alpha/n}$ is
replaced with the fundamental solution $\Phi_\alpha$ of the fractional heat
operator $(\frac{\partial}{\partial t}-\Delta)^{\alpha/2}$.
