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In the work the solvability of nonlocal boundary value problems for thirdorder pseudoparabolic equations in anisotropic spaces of Sobolev is studied. The condition is specified by a spatial variable that combines the generalized Samarsky-Ionkin condition and the integral type condition is particularity of the problems under study. The work aim is to prove the existence and uniqueness of the problems regular solutions - the solutions that have all Sobolev derivatives included in the corresponding equation.

The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

The article is devoted to the study of the solvability of boundary value problems for third-order Sobolev-type differential equations of the third order with two time variables (such equations are also called composite-type equations or equations not solved for the derivative). The peculiarities of the equations under study are, firstly, that the differential operators acting at the time derivatives are not assumed inverse, and, secondly, that the statements of boundary value problems for them are determined by the coefficients of these differential operators. For the problems proposed in the article, we prove existence and uniqueness theorems for regular solutions (solutions having all weak derivatives in the sense of Sobolev involved in the equation). The technique of proving the existence theorems is based on a special regularization of the equations under study, a priori estimates, and passage to the limit.

In this article we investigate the correctness of boundary value problems for a sixth order quasi-hyperbolic equation in the Sobolev space Lu = −D6t u + ∆u − λu (Dt =∂/∂t , ∆ = Σni=1∂2/∂x2i – Laplace operator, λ – real parameter). For the given operator L two spectral problems are introduced and uniqueness of these problems is established. The eigenvalues and eigenfunctions of the first spectral problem are calculated for the sixth order quasi-hyperbolic equation. In this work we show that the equation Lu = 0 for λ < 0 under uniform conditions has a countable set of nontrivial solutions. Usually, this does not happen when the operator L is an ordinary hyperbolic operator.