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

  • J. Alfredo López-Mimbela (a1) and Anton Wakolbinger (a2)

Abstract

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.

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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.

References

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[1] Escobedo, M., and Herrero, M. A. (1991). Boundedness and blowup for a semilinear reaction-diffusion system. J. Diff. Eqns. 89, 176202.
[2] 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, 47100.
[3] Etheridge, A. M. (1996). A probabilistic approach to blowup of a semilinear heat equation. Proc. R. Soc. Edinb. A 126, 12351245.
[4] 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, 105113.
[5] Levine, H. A. (1990). The role of critical exponents in blowup theorems. SIAM Rev. 32, 262288.
[6] Liggett, T. (1985). Interacting Particle Systems. Springer, New York.
[7] 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. 147155.
[8] Lukács, E. (1970). Characteristic Functions, 2nd edn. Griffin, London.
[9] Nagasawa, M., and Sirao, T. (1969). Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Trans. Amer. Math. Soc. 139, 301310.
[10] 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.

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