Vol 60, No 10 (2024)

A heat-electric (1+ 1)-dimensional model of semiconductor heating in an electric field is considered. For the corresponding Cauchy problem, the existence of a classical solution that is short-lived in time is proved, a global a priori estimate is obtained in time, and a result is obtained about the absence of even a classical solution local in time.

The Cauchy problem for a second-order parabolic system with coefficients and the right hand side which belong to the Zygmund anisotropic space is considered. A smoothness scale of the Cauchy problem solutions in anisotropic Zygmund spaces is obtained. A priori estimates of solutions for uniformly elliptic systems in isotropic Zygmund spaces are derived.

For a nonlinear partial differential equation generalizing a damped sixth-order Boussinesq equation with double dispersion and the equation of transverse oscillations of a viscoelastic Voigt–Kelvin beam under the action of external and internal friction and whose deformation is considered taking into account the correction for the inertia of section rotation, sufficient conditions for the existence and exponential decay of a global solution of the Cauchy problem are found.

A multidimensional controllable system with a constant matrix, a significant nonlinearity of the twoposition relay type with hysteresis as a control and a continuous periodic perturbation function is considered. The system matrix has simple, real, non-zero eigenvalues, among which one can be positive. Conditions for the system parameters, including the nonlinearity ones, are established under which there is a single two-point oscillatory periodic solution with a period comparable to the period of the perturbation function in the case of a special type of the feedback vector. The asymptotic stability of the solution has been proven using the phase plane method. The results obtained are illustrated by examples for three-dimensional systems.

For hybrid linear autonomous continuous-discrete systems, methods for designing two types of regulators that provide finite stabilization are proposed. The implementation of one of them, a regulators for finite stabilization by state, is based on knowledge of the values of the control system solution at discrete moments of time, multiples of the quantization step. For this purpose, an observer has been built that makes it possible to obtain the necessary solution values based on the observed output signal in real time and with zero error. The second type of regulator — the regulator of finite stabilization by output — uses the observed output signal as feedback, and its design is a modification of the finite state stabilization regulator by state by including the above observer in its circuit.

The paper is concerned with a functional identity and estimates which are fulfilled for the measures of deviations from exact solutions of the obstacle problem for the 𝑝-Laplacian. They hold true for any functions from the corresponding (energy) functional class, which contains the generalised solution of the problem as well. We do not use any special properties of approximations or numerical methods nor information of the exact configuration of the coincidence set. The right-hand side of the identities and estimates contains only known functions and can be explicitly calculated, and the left side represents a certain measure of the deviation of the approximate solution from the exact solution. The right-hand side of the identity and estimates contains only known functions and can be explicitly calculated, while and the left side represents a certain measure of the deviation of the approximate solution from the exact one. The obtained functional relations allow to estimate the error of of any approximate solutions of the problem regardless of the method of their obtaining. In addition, they enable to compare the exact solutions of problems with different data. The latter provides the possibility to estimate the errors of mathematical models.

The weak solvability of the initial-boundary value problem describing the motion of weakly concentrated aqueous polymer solutions taking into account the memory of the fluid is considered in the paper. In this model the memory is considered along the trajectory of fluid particles, determined by the velocity field. The topological approximation approach and the theory of regular Lagrangian flows are used.

In a real reflexive Banach space, an equation with a parameter and a discontinuous nonlinear operator is considered. Both parameter estimations and operator norms are found for the equation. These estimations validate and define concretely the similar estimations obtained earlier in problems with a parameter for elliptic and ordinary differential equations with discontinuous right-hand sides.