Nonexistence of Positive Solutions to an Elliptic System and Blow-Up Rate for a Parabolic System

We first get the conditions under which the elliptic system -Delta u(1) = u(i+1)(pi), u(s+1) := u(1) (i = 1,2, ..., s) has no positive radially symmetric solutions. Then by using this nonexistence result, we establish blow-up estimates for semilinear reactiondiffusion system u(it) = Delta u(i) + u(i+1)(Pi), u(s+1) := u(1) (i = 1,2, ..., s) with null Dirichlet boundary conditions. The results of our paper with the those in Wang (Comp and Math with Appl, 44, 573-585, 2002) are same, but our methods of proofs are entirely different, even easier than that used.