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

arXiv comment40 pages