Pointwise bounds and blow-up for systems of semilinear parabolic inequalities and nonlinear heat potential estimates

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 0where λ and σ are nonnegative constants and Ω is an open subset of R n , 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.