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

Abstract

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