We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

In the work we study differential operators on arbitrary geometric graphs without loops. We extend the known results for differential operators on an interval to the differential operators on the graphs. In order to define properly the maximal operator on a given graph, we need to choose a set of boundary vertices. The non-boundary vertices are called interior vertices. We stress that the maximal operator on a graph is determined not only by the given differential expressions on the edges, but also by the Kirchhoff conditions at the interior vertices of the graph. For the introduced maximal operator we prove an analogue of the Lagrange formula. We provide an algorithm for constructing adjoint boundary forms for an arbitrary set of boundary conditions. In the conclusion of the paper, we present a complete description of all self-adjoint restrictions of the maximal operator.

We consider the Cauchy problem for a certain class of anisotropic parabolic second-order equations with double non-power nonlinearities. The equation contains an "inhomogeneity" in the form of a non-divergent term depending on the sought function and spatial variables. Non-linearities are characterized by N -functions, for which 2 -condition is not imposed. The uniqueness of renormalized solutions in Sobolev-Orlich spases is proved by the S.N.Kruzhkov method of doubling the variables.

In this paper we consider the polynomial approximate solutions of the Dirichlet problem for minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation.

By the method of monotone operators we establish theorems on global existence and uniqueness, as well as estimats and methods of finding the solutions for various classes of nonlinear convolution type integral equations in the real space of 2periodic functions (-, ).

Given an action of a discrete group on a smooth compact manifold with a boundary, we consider a class of operators generated by pseudodifferential operators on and shift operators associated with the group action. For elliptic operators in this class, we obtain a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the -group of the crossed product of the algebra of continuous functions on the cotangent bundle over the interior of the manifold and the group acting on this algebra by automorphisms.

We consider a hydrodynamic type system, waterbag model, that admits a dispersionless Lax representation with a logarithmic Lax function. Using the Lax representation, we construct a recursion operator of the system. We note that the constructed recursion operator is not compatible with the natural Hamiltonian representation of the system.

The paper is devoted to Cauchy problem for equation pq pq 0, where pq is a -periodic function. It is known that for a wide class of the nonlinearities pq the "most part" of solutions of Cauchy problem for this equation are singular, i.e., they tend to infinity at some finite point of the real axis. Earlier in the case pq 3 this fact allowed us to propose an approach for a complete description of solutions to this equation bounded on R. One of the ingredients in this approach is the studying of the set introduced as the set of the points p , I q in the initial data plane, for which the solutions to the Cauchy problem p0q , p0q I are not singular in the segment r0; s. In the present work we prove a series of statements on the set and on their base, we classify all possible type of the geometry of such sets. The presented results of the numerical calculations are in a good agreement with theoretical statements.

We introduce modulo-loxodromic functions and study their representations, zeroes and poles distribution. We also show that each modulo-loxodromic meromorphic function in C {0} is Julia exceptional.