A critical exponent of Fujita type for a nonlinear reaction–diffusion system on a Riemannian manifold

In this paper, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction–diffusion system { u t − Δ u = W ( x ) v p + S ( x ) in  M n × ( 0 , ∞ ) , v t − Δ v = F ( x ) u d + G ( x ) 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, and S ( x ) , G ( x ) are non-negative L l o c 1 functions. We assume that both u 0 ( x ) and v 0 ( x ) are non-negative, smooth and bounded functions, and constants p , d > 1 . When p = d , there is an exponent p ∗ which is critical in the following sense. When p ∈ ( 1 , p ∗ ] , the above problem has no global positive solution for any non-negative constants S ( x ) , G ( x ) not identically zero. When p ∈ [ p ∗ , ∞ ) , the problem has a global positive solution for some S ( x ) , G ( x ) > 0 and u 0 ( x ) , v 0 ( x ) ≥ 0 .