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 C2,1 nonnegative solutions u(x,t) and v(x,t) of the system0≤ut−Δu≤vλ0≤vt−Δv≤uσ in Ω×(0,1), where λ and σ are nonnegative constants and Ω is an open subset of Rn, n≥1. We provide optimal conditions on λ and σ such that solutions of this system satisfy pointwise bounds in compact subsets of Ω as t→0+. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.