Non-existence of global solutions to systems of non-autonomous nonlinear parabolic equations

We consider the non-autonomous system of nonlinear parabolic equations { u_t+ t^rΔ_αu = v^q v _t + t^sΔ_βv=u^p posed in Q := (0, ∞) × R^N, subject to the initial data (u(0,x) = u₀(x), v(0, x) = vo(x)), where p > 1 and q > 1 are positive real numbers, α, β €]0,2] and Δ_γ := (-Δ)^γ/2 is the (-Δ)^γ/2 fractional power of -Δ in the x variable defined via the Fourier transform and its inverse ^-1 by (-Δ)^γ/2ω(x,t) = ^-1 (v^γ(ω) (ξ)) (x,t), where r > -1 and s > -1. The Fujita critical exponent which separates the case of blowing-up solutions from the case of globally in time existing solutions is determined.