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

By lower estimates of the functionals 𝔼[e S t K t N t ], 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 ∂ w t /∂ t = A w t + V w t β . Roughly speaking, the growth of the tree length has to win against the ‘mobility’ of the motion belonging to the generator A , since, in the probabilistic representation of the equations, the latter results in small K ( t ) as t → ∞. In the single-type situation, this gives a re-interpretation of classical results of Nagasawa and Sirao(1969); in the multitype scenario, part of the results obtained through analytic methods in Escobedo and Herrero (1991) and (1995) are re-proved and extended from the case A = Δ to the case of α-Laplacians.