Initial traces and solvability of Cauchy problem to a semilinear parabolic system

Abstract

Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_t u = D_1 \Delta u+v^p \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ \partial_t v = D_2 \Delta v+u^q \quad \mbox{in}\quad \mathbf{R}^N \times (0,T),\\ u,v \ge 0 \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ (u(\cdot,0),v(\cdot,0)) = (\mu,\nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right. $$ 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 Radon measures or nonnegative measurable functions in $\mathbf{R}^N$. In this paper we study qualitative properties of the initial trace of the solution $(u, v)$ and obtain necessary conditions on the initial data $(\mu, \nu)$ for the existence of solutions to problem (P).