Summary: We consider the weakly coupled system of reaction-diffusion equations
where , , with and , . Put
and let and be the spaces of nonnegative, bounded continuous functions satisfying
respectively. At , initial values are prescribed. It is proved that if or if with or with , then every nontrivial nonnegative solution is not global in time; whereas if and with , , then there exist both global solutions and nonglobal solutions. Moreover, we obtain the asymptotic behavior as of the global solutions.