Transition from decay to blow-up in a parabolic system

Pavol Quittner

Address. Institute of Applied Mathematics, Comenius University, Mlynska dolina, SK-84215 Bratislava, Slovakia

E-mail: quittner@fmph.uniba.sk

Abstract. 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 $a>0$, $b\geq0$ and $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\leq2$. In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded.

AMS classification. 35K60, 35J65, 35B40

Key words. Blow-up, global existence, apriori estimates