Initial pointwise bounds and blow-up for parabolic Choquard-Pekar inequalities

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 $\sigma \ge 0$ are constants, $\mathrm{\Phi }$ is the heat kernel, and $\ast$ 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 ${\mathrm{\Phi }}^{\alpha \mathrm{/}n}$ is replaced with the fundamental solution ${\mathrm{\Phi }}_{\alpha }$ of the fractional heat operator $(\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$.