Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T04:29:17.124Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Partial differential equations Markov processes Parabolic equations and systems

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
∗∗Postal address: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 R1,this answers a question raised in [4].

Keywords

Branching particle systemYule treesemilinear partial differential equationglobal solution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 35K57: Reaction-diffusion equations 35B35: Stability
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 635 - 641
Copyright
Copyright © Applied Probability Trust 2000 

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 Levine, H. (1995). Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Ration. Mech. Anal. 129, 47–100.Google Scholar
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, 109–124.Google Scholar
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, 147–155.Google Scholar
López-Mimbela, J. A., and Wakolbinger, A. (1998). Length of Galton–Watson trees and blow-up of semilinear systems. J. Appl. Prob. 35, 802–811.CrossRefGoogle Scholar
McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28, 323–331.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