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Nonexplosion of a class of semilinear equations via branching particle representations
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- Journal:
- Advances in Applied Probability / Volume 40 / Issue 1 / March 2008
- Published online by Cambridge University Press:
- 01 July 2016, pp. 250-272
- Print publication:
- March 2008
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We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3, …. The corresponding branching process is related to the semilinear partial differential equation
for x ∈ ℝd, where A is the infinitesimal generator of a multiplicative semigroup and the pks, k = 2, 3, …, are nonnegative functions such that 
We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.
for 
We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.A probabilistic proof of non-explosion of a non-linear PDE system
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- Journal:
- Journal of Applied Probability / Volume 37 / Issue 3 / September 2000
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- 14 July 2016, pp. 635-641
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- September 2000
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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].
remain bounded for suitable bounded initial conditions, provided 