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

Abstract

Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_1 \Delta u+v^{p} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ \partial_{t} v = D_2 \Delta v+u^{q} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ u, v \geq 0 \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ (u(\cdot, 0), v(\cdot, 0)) = (\mu, \nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right. $$ where $N \geq 1$, $T > 0$, $D_1 > 0$, $D_2 > 0$, $0 < p \leq q$ 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).