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A probabilistic proof of non-explosion of a non-linear PDE system

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

Abstract

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R 1,this answers a question raised in [4].

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Postal address: Apartado Postal 402, Guanajuato 36000, Mexico
∗∗ Postal address: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany. Email address: wakolbin@math.uni-frankfurt.de

References

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[1] Escobedo, M., and Levine, H. (1995). Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Ration. Mech. Anal. 129, 47100.
[2] Fujita, H. (1966). On the blowing up of solutions of the Cauchy problem for u_t=δ u + u1+α . J. Fac. Sci. Univ. Tokyo Sect. I 13, 109124.
[3] López-Mimbela, J. A. (1996). A probabilistic approach to existence of global solutions of a system of nonlinear differential equations. Aportaciones Matemáticas Notas de Investigación 12, 147155.
[4] López-Mimbela, J. A., and Wakolbinger, A. (1998). Length of Galton–Watson trees and blow-up of semilinear systems. J. Appl. Prob. 35, 802811.
[5] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28, 323331.
[6] 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.

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