Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities

We study the behavior as $t \to 0^+$ of nonnegative functions

$u\in {{C}^{2,1}}({{\mathbb{R}}^{n}}\times (0,1))\cap {{L}^{\lambda }}({{\mathbb{R}}^{n}}\times (0,1)),\ \ n\ge 1,\ \ \ \ \ \ \ \ \ \ \ \ \left( 0.1 \right)$

satisfying the parabolic Choquard-Pekar type inequalities

$0\le {{u}_{t}}-\Delta u\le ({{\Phi }^{\alpha /n}}*{{u}^{\lambda }}){{u}^{\sigma }}\ \ \text{ in }{{B}_{1}}\ (0)\times (0,1)\ \ \ \ \ \ \ \ \ \ \left( 0.2 \right)$

where $α∈(0, n+2)$, $λ>0$, and $σ≥0$ are constants, $Φ$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n× (0, 1)$. We provide optimal conditions on $α, λ$, and $σ$ such that nonnegative solutions $u$ of (0.1), (0.2) satisfy pointwise bounds in compact subsets of $B_1(0)$ as $t \to0^+$. We obtain similar results for nonnegative solutions of (0.1), (0.2) when $Φ^{α/n}$ in (0.2) is replaced with the fundamental solution $Φ_α$ of the fractional heat operator $(\frac{\partial}{\partial t}-Δ)^{α/2}$.