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This paper deals with the global existence and blow-up of positive solutions to the systems: u_t = ∇(u^∇u) + u¹ + v^a v_t = ∇(v^n∇v) + u^b + v^k in B_R × (0, T) ∂u∂η = u^αv^p, ∂v∂η = u^qv^β on S_R × (0, T) u(x, 0) = u_0(x), v(x, 0