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A Probabilistic Proof of Non-Explosion of a Non-Linear PDE System

J. Alfredo López-Mimbela and Anton Wakolbinger
Journal of Applied Probability
Vol. 37, No. 3 (Sep., 2000), pp. 635-641
Stable URL: http://www.jstor.org/stable/3215601
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Probabilistic Proof of Non-Explosion of a Non-Linear PDE System
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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 u̇ = Au+uv, v̇ = Bv+uv 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].

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