Pointwise Bounds and Blow-up for Systems of Semilinear Parabolic Inequalities and Nonlinear Heat Potential Estimates

TL;DR

This paper establishes optimal conditions on parameters for solutions of a coupled parabolic system to remain bounded near initial time, using new heat potential bounds as a key tool.

Contribution

It introduces novel pointwise bounds for nonlinear heat potentials and determines conditions ensuring boundedness of solutions as time approaches zero.

Findings

Derived optimal conditions on  and  for solution bounds

Established new bounds for nonlinear heat potentials

Applied bounds to analyze solution blow-up behavior

Abstract

We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-\Delta u\leq v^\lambda\] \[0\leq v_t-\Delta v\leq u^\sigma\] in $\Omega\times (0,1)$, where $\lambda$ and $\sigma$ are nonnegative constants and $\Omega$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $\lambda$ and $\sigma$ such that solutions of this system satisfy pointwise bounds in compact subsets of $\Omega$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.