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

TL;DR

This paper investigates the initial behavior and bounds of nonnegative solutions to parabolic Choquard-Pekar inequalities, establishing optimal conditions for their pointwise bounds as time approaches zero.

Contribution

It provides the first comprehensive analysis of initial pointwise bounds for solutions to these inequalities, including fractional heat operator cases, under optimal conditions.

Findings

Derived optimal conditions for solution bounds as t→0+

Extended results to fractional heat operators

Identified critical parameters for solution behavior

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