Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T13:20:19.662Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

J. Alfredo López-Mimbela  and
Anton Wakolbinger
J. Alfredo López-Mimbela*
Affiliation:
Centro de Investigación en Matemáticas
Anton Wakolbinger*
Affiliation:
J. W. Goethe-Universität Frankfurt am Main
*
∗Postal address: Apartado Postal 402, Guanajuato 36000, Mexico. Email address: jalfredo@fractal.cimat.mx
∗∗Postal address: FB Mathematik (12), J. W. Goethe Universität, D-60054 Frankfurt am Main, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

By lower estimates of the functionals 𝔼[eStKtNt], where St and Nt 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 ∂wt/∂t = Awt + Vwtβ. 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.

Keywords

Galton–Watson treesmultitype branching particle systemssemilinear partial differential equations

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 802 - 811
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Escobedo, M., and Herrero, M. A. (1991). Boundedness and blowup for a semilinear reaction-diffusion system. J. Diff. Eqns. 89, 176–202.Google Scholar
Escobedo, M., and Levine, H. (1995). Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Rational Mech. Anal. 129, 47–100.Google Scholar
Etheridge, A. M. (1996). A probabilistic approach to blowup of a semilinear heat equation. Proc. R. Soc. Edinb. A 126, 1235–1245.Google Scholar
Fujita, H. (1966). On the blowing up of solutions to the Cauchy problem for ut = Δu + u 1+α . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 16, 105–113.Google Scholar
Levine, H. A. (1990). The role of critical exponents in blowup theorems. SIAM Rev. 32, 262–288.Google Scholar
Liggett, T. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
López-Mimbela, J. A. (1996). A probabilistic approach to existence of global solutions of a system of nonlinear differential equations. In Aportaciones Matemáticas, Serie Notas de Investigación 12, Sociedad Matemática Mexicana, pp. 147–155.Google Scholar
Lukács, E. (1970). Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
Nagasawa, M., and Sirao, T. (1969). Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Trans. Amer. Math. Soc. 139, 301–310.Google Scholar
Velázquez, J. J. L. (1994). Blow up for semilinear parabolic equations. In Recent Advances in Partial Differential Equations, ed. Herrero, M. A. and Zuazua, E. Wiley, New York.Google Scholar