Transition from decay to blow-up in a parabolic system

summary:We show a locally uniform bound for global nonnegative solutions of the system $u_t=\Delta u+uv-bu$, $v_t=\Delta v+au$ in $(0,+\infty )\times \Omega $, $u=v=0$ on $(0,+\infty )\times \partial \Omega $, where $a>0$, $b\ge 0$ and $\Omega $ is a bounded domain in $\mathbb {R}^n$, $n\le 2$. In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded