Existence of solutions for a semilinear parabolic system with singular initial data

Abstract

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2Δv+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$, and $(\mu ,\nu )$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb{R}}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.