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Vol 60, No 10 (2024)
PARTIAL DERIVATIVE EQUATIONS
THE CAUCHY PROBLEM FOR AN NONLINEAR WAVE EQUATION
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1299-1311
1299-1311
INITIAL PROBLEMS FOR THE ABSTRACT LEGENDRE EQUATION CONTAINING TWO PARAMETERS
Abstract
Using the concept of a fractional integral of a function over another function, transformation operators are constructed that make it possible to prove the solvability of initial problems for the abstract singular Legendre equation containing two parameters. Examples are given.
Differencial'nye uravneniya. 2024;60(10):1312-1324
1312-1324
HE CAUCHY PROBLEM FOR PARABOLIC SYSTEM WITH VARIABLE COEFFICIENTS IN ANISOTROPIC ZYGMUND SPACES
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1325-1333
1325-1333
ON EXACT SOLUTIONS OF MULTIDIMENSIONAL GENERALIZED MONGE–AMPERE EQUATION
Abstract
Exact solutions of some multidimensional generalized Monge–Ampere equations are found. These solutions are a superposition of a quadratic form of spatial variables and solutions of nonlinear ordinary differential equations generated by the Monge–Ampere equations.
Differencial'nye uravneniya. 2024;60(10):1334-1349
1334-1349
ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A NONLINEAR EQUATION
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1350-1367
1350-1367
CONTROL THEORY
CONTROL DESIGN FOR A MULTIDIMENSIONAL SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS WITH RELAY HYSTERESIS AND PERTURBATION
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1368-1385
1368-1385
1386-1393
REGULATORS OF FINITE STABILIZATION FOR HYBRID LINEAR CONTINUOUS-DISCRETE SYSTEMS
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1394-1406
1394-1406
NUMERICAL METHODS
A POSTERIORI ERROR ESTIMATES FOR APPROXIMATE SOLUTIONS OF THE OBSTACLE PROBLEM FOR THE 𝑝-LAPLACIAN
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1407-1421
1407-1421
BRIEF MESSAGES
ON WEAK SOLVABILITY OF MATHEMATICAL MODEL DESCRIBING THE MOTION OF POLYMER SOLUTIONS WITH MEMORY
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1422-1428
1422-1428
1429-1434
ON ESTIMATIONS IN AN EQUATION WITH A PARAMETER AND A DISCONTINUOUS OPERATOR
Abstract
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.
Differencial'nye uravneniya. 2024;60(10):1435-1440
1435-1440