We study the non-local reaction-diffusion system with Neumann boundary conditionswhere p, q > 0, u 0 (x), v 0 (x) ∈ C( ) are nonnegative and nontrivial functions, ∈ R N a bounded connected and smooth domain. We determine the existence and uniqueness of the solution. The blow-up phenomenon is considered and the blow-up rates are obtained.