Optimal singularities of initial functions for solvability of a semilinear parabolic system

Abstract

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in {\mathbf{R}}^N,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in {\mathbf{R}}^N,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in {\mathbf{R}}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(\mu ,\nu )$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\mathbf{R}}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.