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

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

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: wakolbin@math.uni-frankfurt.de
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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 

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References

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