Vol 243 (2025)

In this paper, we examine a one-dimensional reaction-diffusion system with different-scale diffusion coefficients, discontinuous reaction functions, and Neumann boundary conditions. We demonstrate that a singular perturbation in the fast-component equation and reaction discontinuities lead to the formation of contrast structures with internal transition layers. Also, we analyze the existence, uniqueness, and asymptotic stability of stationary solutions. The obtained results provide theoretical justification for numerical methods applicable to such systems and enable prediction of behavior of solutions in domains of sharp gradients, which is crucial for developing efficient computational algorithms.

The process of the formation of a front-type solution in the reaction-diffusion equation in the case of spatially inhomogeneous nonlinear diffusion is considered. Sufficient conditions are formulated to ensure the formation of a sharp inner transition layer (front) in a neighborhood of a certain point, and estimates of the duration of the transition process are obtained.

This paper is devoted to the study of the concept of normality in real AW*-algebras. The authors examine whether normality is preserved when passing from a complex AW*-algebra to its real part and prove that all real AW*-factors are normal. Conditions are established under which a real AW*-algebra is normal, in particular, when its center is locally $\sigma$-finite. The obtained results serve as real analogs of known theorems on normality in AW*-algebras and contribute to the development of operator algebra theory in the real setting.

In this paper, we consider the problems of synthesis of positional control for linear dynamic systems in the case where it is necessary to guarantee the achievement of the control goal, and the disturbances acting on the dynamic system and the interference in the information channels of the system are known with an accuracy of sets in which they can take any values. We construct information sets and forecast sets that contain the state vector. Control problems are solved for the case of specifying requirements for the system in the form of sets in the phase space to which the state vector must belong, taking into account the constraints on control, or with requirements in the form of a quadratic functional. The application of Lyapunov functions for control synthesis is shown. The first part of this work is devoted to evaluation algorithms.