We consider the parabolic system ut−Δu=f(t)urvs,vt−Δv=g(t)uqvs, in Ω×(0,T), where Ω⊂RN is either an unbounded or bounded domain and f,g∈C[0,∞). We find conditions for the global existence or nonglobal existence, which are expressed in terms of the behavior of ‖S(t)u0‖∞ as t→∞, where u(t)=S(t)u0 is the solution of the linear problem ut−Δu=0,u(0)=u0.