The reaction diffusion system with anomalous diffusion and a balance law ut+-Δα/2u=-fu,v, vt+-∆β/2v=fu,v, 0<α, β<2, is con sidered. The existence of global solutions is proved in two situations: (i) a polynomial growth condition is imposed on the reaction term f when 0<α≤β≤2; (ii) no growth condition is imposed on the reaction term f when 0<β≤α≤2
