Blow-up for a system with time-dependent generators

Sufficient conditions, for finite time blow-up of nonnegative solutions, are given for the weakly coupled system: partial derivative u(1)/partial derivative t = k(1) (t) Delta(alpha 1)u(1) + u(1)u(2)(beta 1), u(1) (0) = phi(1), partial derivative u(2)/partial derivative t = k(2) (t) Delta(alpha 2)u(2) + u(2)u(1)(beta 2), u(2) (0) = phi(2), where Delta(alpha i) is the fractional Laplacian, 0 < alpha(i) <= 2, beta(i) >= 1 are constants, phi(i) are positive, bounded, continuous and not identically zero, and k(i) : [0, infinity) -> [0, infinity), are continuous with integral(t)(0) k(i) (r) dr = O (t(rho i)), rho(i) > 0, for i = 1, 2.