The aim of this paper is to
investigate the existence and nonexistence of a nonnegative global solution for a
semilinear parabolic system in a bounded domain. It is shown that for a certain class
of initial functions the corresponding solution of the initial boundary value problem
has a finite escape time, while for another class of initial functions a unique solution
exists for all time and diminishes to zero. This result leads to an explicit estimate for
the stability and the instability regions of the trivial steady-state solution. In the case
of nonexistence of global solutions, an estimate for the finite escape time is also
given.
