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 \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] 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.

arXiv comment31 pages, one figure