Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion

In this article we study the positive solutions of the parabolic semilinear system of competitive type \left\{\begin{array} [c]{c}% u_{t}-\Delta u+v^{p}=0, v_{t}-\Delta v+u^{q}=0, \end{array} \right. in $\mathrm{\Omega }\times (0,T)$, where $\mathrm{\Omega }$ is a domain of ${\mathbb{R}}^N,$ and $p,q>0,$ $pq\mathrm{\ne }1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form $$u(x,t)\leqq C{t}^{-(p+1)\mathrm{/}(pq-1)},v(x,t)\leqq C{t}^{-(q+1)\mathrm{/}(pq-1)}$$ in $\omega \times (0,T_1),$ for any domain $\omega \subset \subset \mathrm{\Omega }$ and $T_1\in (0,T),$ and $C=C(N,p,q,T_1$ For $p,q>1,$ we prove the existence of an initial trace at time 0, which is a Borel measure on $\mathrm{\Omega }\mathrm{.}$ Finally we prove that the punctual singularities at time $0$ are removable when $p,q\geqq1+2/N.