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 $\sigma\geq0$ 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 $(\frac{\partial}{\partial t}-\Delta)^{\alpha/2}$.