Length of Galton-Watson trees and blow-up of semilinear systems

By lower estimates of the functionals E[e(St) K(t)(Nt)], where S-t and N-t denote the total length up to time t and the number of individuals at time t in a Galton-Watson tree, we obtain sufficient criteria for the blow-up of semilinear equations and systems of the type partial derivative w(t)/parti al derivative t = Aw(t) + Vw(t)(beta). Roughly speaking, the growth of the tree length has to win against the 'mobility' of the motion belonging to th e generator A, since, in the probabilistic representation of the equations, the latter results in small K (t) as t --> infinity. Ln the single-type si tuation, this gives a re-interpretation of classical results of Nagasawa an d Sirao ([9]), in the multitype scenario, part of the results obtained thro ugh analytic methods in [1] and [2] are re-proved and extended from the cas e A = Delta to the case of alpha-Laplacians.