For the absolutely convergent in a half-plane Dirichlet series we establish upper estimates without exceptional sets.

Using the method of characteristics and the Banach fixed point theorem we established the existence and uniqueness of a global classical (smooth) solution to an initial-boundary value problem with nonlocal boundary conditions for a hyperbolic integro-differential system involving equations without time derivative of unknown functions.

We consider a quasilinear parabolic boundary-value problem in a two-level thick junction Ω ε of type 3:2:2 , which is the union of a cylinder Ω 0 and a large number of ε -periodically situated thin discs with variable thickness. Different Robin boundary conditions with perturbed parameters are given on the surfaces of the thin discs. The leading terms of the asymptotic expansion are constructed and the corresponding estimate in Sobolev space is obtained.

A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform topo- logical space.We introduce three new constructions (namely, a ballean-filter mix, a ballean-ideal mix and a filter product of directed sets) to give some balleans with extremal properties. In particular, we construct a non-metrizable Frechet group ballean.

We consider boundary value problems for infinite triangular systems of elliptic equations with variable coefficients in 3d Lipschitz domains. Variational formulations of Dirichlet, Neumann and Robin problems are received and their well posedness in corresponding Sobolev spaces is established. With the help of introduced q-convolution the integral representations of generalized solutions of formulated problems in the case of constant coefficients are built. We investigate the properties of integral operators and well posedness of received systems of boundary integral equations.

Consider a mixed problem for a class of system of a high order doubly nonlinear parabolic equations with variable exponent of nonlinearity. This problem is considered in generalized Lebesgue-Sobolev spaces. As a result, we reached a condition of the existence of a solution. We use here Galerkin's procedure.

We study the M θ /G/1/m queue with the time of service, depending on the length of the queue at the service initiation. By using Korolyuk's potential method, we derive the average duration of the busy time and the stationary distribution of the number of customers in such a system. Similar results for the M θ /G/1/m queue with one threshold of switching of service modes are obtained.

In the paper we consider entire functions f:C p →C, p≥2, defined by power series f(z)=f(z 1 ,…,z p )=∑ +∞ ∥n∥=0 a n z n ,z n =z n 1 1 ⋅…⋅z n p p , n=(n 1 ,…,n p ). For r=(r 1 ,…,r p )∈R p + we set M f (r)=max{|f(z)|:|z i |≤r i ,i∈{1,…,p}}, μ f (r)=max{|a n |r n :n∈Z p + }, r ∨ =max{r i :i∈{1,…,p}}, r ∧ =min{r i :i∈{1,…,p}} and let l be a log-convex real function on (1,+∞) such that lnt=o(l(t)), t→+∞. Then for any entire transcendental function f {with} lnM f (r)≤l(r ∨ ), r ∧ →+∞, {the} inequality lim ¯ ¯ ¯ ¯ ¯ r ∧ →+∞ lnM f (r)−lnμ f (r) lnlnμ f (r) ≤α holds if and only if lim ¯ ¯ ¯ ¯ ¯ t→+∞ (lnl(t)/lnlnt)≤1+α/p. Similar theorems are proved for random entire functions of several complex variables.