In this paper, we study the global existence and nonexistence of solutions to the following semilinear parabolic system { u t − Δ u = v p in M n × ( 0 , ∞ ) , v t − Δ v = u q in M n × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) in M n , v ( x , 0 ) = v 0 ( x ) in M n , where M n ( n ≥ 3 ) is a non-compact complete Riemannian manifold, Δ is the Laplace–Beltrami operator. We assume that both u 0 ( x ) and v 0 ( x ) are nonnegative, smooth and bounded functions, constants p , q ≥ 1 .