Vol 61, No 8 (2025)

On a Riemannian manifold of dimension m, m ≥ 3, we consider a non-empty compact set A that is invariant for some C^1-smooth flow. Sufficient conditions are proposed under which A is a hyperbolic set of the flow.

It is shown that a periodic 𝐸-function whose derivatives at zero are integer algebraic numbers satisfies a differential equation of the form 𝑃(𝑦, 𝑦′) = 0 where 𝑃 is a polynomial with algebraic coefficients. Consequently, it is proven that every such function is a Laurent polynomial in an exponential 𝑒𝛼𝑧.

The boundedness of the constructed biharmonic function in an unbounded three-dimensional domain is proved if this function and its normal derivative are bounded on the boundary of this domain.

We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape minimizing the weight at a given fundamental frequency of shell vibrations. Using the obtained formula for the gradients of the components of the eigenfunction corresponding to the minimum eigenvalue, the second Frechet differentiability of the frequency functional is established. It is proved that when the necessary conditions are met, the sufficient conditions are also realized.

Time-varying linear and nonlinear descriptor systems of observation with discrete time are considered. In the linear case, the criteria for observability on the finite horizon are obtained, and the conditions for robust observability are found. The duality theorems linking the properties of controllability and observability are proved. For nonlinear systems, the conditions of local observability on the finite horizon are obtained, using linear approximation.

Compact and monotone difference schemes of the fourth order of accuracy, preserving the property of conservatism (divergence) for the one-dimensional and two-dimensional quasilinear stationary reaction-diffusion equation are constructed and investigated. A priori estimates of the difference solution in the nonlinear case for the one-dimensional quasilinear equation are obtained based on the established two-sided estimates of the grid solution. For the linearization of the nonlinear difference scheme, an iterative method of the Newton-Seidel type is used, preserving conservatism and monotonicity. The main idea of the proposed difference schemes is based on the possibility of parallelizing the computational process. The emerging problems of finding additional boundary conditions at boundary nodes in both one-dimensional and two-dimensional cases are solved using the Newton interpolation polynomial of the fourth order of accuracy. The presented results of the computational experiments illustrate the increased order of the proposed algorithms. The possibility of generalizing this method to non-stationary quasilinear equations is also indicated.

A boundary value problem for the Helmholtz equation in a circle is considered, with the boundary condition containing an oblique derivative with Hölder coefficients. The solvability of the problem in a unique way is proved under certain restrictions on the parameter.

Ниже публикуются краткие аннотации докладов, состоявшихся в весеннем семестре 2025 г. (предыдущее сообщение о работе семинара дано в журнале “Дифференциальные уравнения”. 2025. Т. 61. № 2; дополнительная информация по адресу iline@cs.msu.ru