A nonlocal in time parabolic system whose Fujita critical exponent is not given by scaling

We consider the nonlocal coupled parabolic systemut−Δu=∫0t(t−s)−γ1|v|p−1v(s)ds,vt−Δv=∫0t(t−s)−γ2|u|q−1u(s)ds in (0,T)×Ω(0,T)×Ω when either Ω=RNΩ=RN or Ω⊂RNΩ⊂RN is a bounded domain with smooth boundary. We assume that 0⩽γ1,γ2<10⩽γ1,γ2<1, p,q⩾1p,q⩾1 and the initial data u(0),v(0)∈C0(Ω)u(0),v(0)∈C0(Ω). We obtain the Fujita critical exponent for the system above which is not given by the scaling argument.