We get simple conditions under which the elliptic system −Δu=up1vq1, −Δv=up2vq2inRn,n≥3 withpi+qi>1,i=1,2 has no positive radially symmetric solutions. Then by using this nonexistence result, we establish blow-up estimates for reaction-diffusion system of Fujita typeut=Δu+up1vq1,vt=Δv+up2vq2with the homogeneous Dirichlet boundary value condition.