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

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 $$0\le u_t-\mathrm{\Delta }u\le v^{\lambda }$$ $$0\le v_t-\mathrm{\Delta }v\le u^{\sigma }$$ in $\mathrm{\Omega }\times (0,1)$, where $\lambda$ and $\sigma$ are nonnegative constants and $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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.