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

Abstract

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^λ(\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-Δu\leq(Φ^{α/n}*u^λ)u^σ\quad \text{ in }B_1 (0)\times (0,1) \end{equation} where $α\in(0,n+2)$, $λ>0$, and $σ\geq0$ are constants, $Φ$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal conditions on $α,λ$, and $σ$ 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 $Φ^{α/n}$ is replaced with the fundamental solution $Φ_α$ of the fractional heat operator $(\frac{\partial}{\partial t}-Δ)^{α/2}$.