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

We consider the nonlocal coupled parabolic system u t − Δ u = ∫ 0 t ( t − s ) − γ 1 | v | p − 1 v ( s ) d s , v t − Δ v = ∫ 0 t ( t − s ) − γ 2 | u | q − 1 u ( s ) d s in ( 0 , T ) × Ω when either Ω = R N or Ω ⊂ R N is a bounded domain with smooth boundary. We assume that 0 ⩽ γ 1 , γ 2 < 1 , p , q ⩾ 1 and the initial data u ( 0 ) , v ( 0 ) ∈ C 0 ( Ω ) . We obtain the Fujita critical exponent for the system above which is not given by the scaling argument.