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

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ v D_{2} u^{q}, (u(\cdot,0), v(\cdot,0)) (\mu, \nu), \end{array} \right. where $D_{1}$, $D_{2} 0$, $0 < p \leq q$ with $pq 1$ and $(\mu, \nu)$ is pair of Radon measures or measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on initial data for solvability problem (P) clarify optimal singularities solvability.