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

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