Blow-up and global existence for semilinear parabolic systems with space-time forcing terms

We investigate the finite time blow-up and global existence of sign-changing solutions to the Cauchy problem for the inhomogeneous semilinear parabolic system with space-time forcing terms{ut−Δu=|v|p+tσw1(x),x∈RN,t>0,vt−Δv=|u|q+tγw2(x),x∈RN,t>0,(u(0,x),v(0,x))=(u0(x),v0(x)),x∈RN,where N≥1,p,q>1,σ,γ>−1,σ,γ≠0,w1,w2≢0, and u0,v0∈C0(RN). For the finite time blow-up, two cases are discussed under the conditions wi∈L1(RN) and ∫RNwi(x)dx>0,i=1,2. Namely, if σ>0 or γ>0, we show that the (mild) solution (u,v) to the considered system blows up in finite time, while if σ,γ∈(−1,0), then a finite time blow-up occurs when N2σγ and q>γσ, we show that the solution is global for suitable initial values and wi,i=1,2.