## 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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