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Transition from decay to blow-up in a parabolic system

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Pavol Quittner
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** 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