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Asymptotics of the solution of a mixed problem for a system of differential equations connected to a positive integral operator. (Russian) Zbl 0655.60051

The author considers a mixed boundary value problem for the following system of hyperbolic-type partial differential equations \[ \partial u_ i(x,t)/\partial t+\delta \quad ib_ i(x)\partial u_ i(x,t)/\partial x- \sum^{N}_{k=1}a\quad k_ i(x)u\quad_ k(x,t)=f_ i(x,t),\quad x\in [a,b],\quad t\geq 0,\quad i=1,2,...,N. \] Utilizing both the connection of this system with the functionals of branching processes with a finite number of particles and the Laplace transform the asymptotic behavior of the solutions \(\{u_ i(x,t)\}\) is obtained for \(t\to \infty\). Particularly the first term of the asymptotic expansion is written out explicitly.
Reviewer: A.Dorogovtsev

MSC:

60H99 Stochastic analysis
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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