We consider a non-local second boundary value problem for a mixed-type equation in a cylindrical domain which reduces to the solvability of a local boundary value problem for a mixed-type integro-differential equation in a weighted Sobolev space. The solvability of the latter is established with the use of the successive approximations method. Under certain conditions, the coefficients of the equation are used to prove a theorem on unambiguous regular solvability of a non-local second boundary value problem in a weighted Sobolev space. The coefficient for the second derivative can change sign at the upper base of a cylindrical domain, with a negative value for the lower base.

A class of systems of nonlinear integro-differential equations is considered and the exponential stability of the zero solution is studied. With the use of a special Lyapunov–Krasovskii functional, we establish stability conditions, estimates for attraction sets, and estimates characterizing decay rates of solutions at infinity.

We study boundary value problems (BVP) for parabolic equations with changing direction of evolution. An essential feature of our equations is a discontinuous and sign changing coefficient with a first-kind discontinuity. Depending on the coefficients, we propose correct BVP formulations and prove existence and uniqueness of regular solutions to these BVPs.

A functional identity is obtained that connects a solution of the Cauchy type problem for a linear equation in a Banach space solved with respect to the Riemann– Liouville fractional derivative with elements of the kernel of the operator, which is conjugate to the operator from an equation. This identity is used to obtain theorems on the unique solvability of an inverse problem of restoring the order of a fractional derivative in an equation. Abstract results are applied to considering some problems for partial differential equations.

This work is a continuation of the research begun in [1]. The conditions for the smoothness of the input data are borderline between those for which the solution (statement) of the Stefan problem is weak (the statement does not include the equation for the phase-change boundary) and those for which it is strong, i.e. classical (the phases are separated by a smooth boundary). The strong global time solvability of the problem is established both in the case of an initial temperature close to the phase transition temperature and without this condition.

The purpose of this article is to introduce the concept of possible implementation of continuum mechanics models based on kinetic theory. The fundamental equation of rarefied gas dynamics – the Boltzmann equation – is an integro-differential equation, extremely complex when its actual resolution is necessary. The mismatch between the rightand left-hand sides of the Boltzmann kinetic equation is well known, especially in numerical experiments. The imperfection of the Boltzmann kinetic equation led to the need to construct so-called discrete kinetic equations. Several simplified models have been proposed, including discrete kinetic equations with a finite number of group velocities. These models have interesting conceptual and mathematical features. Invariant solutions in discrete kinetics are states or relations that are preserved over time during the evolution of a system described by discrete kinetic equations. Their role is fundamental both in theoretical analysis and in applications.

The invariants reflect the fundamental conservation laws for the number of particles (the mass balance), momentum, energy, charge, etc. In discrete models, this ensures that the numerical scheme does not artificially generate or destroy physical quantities. We consider the problem of closing moment chains constructed with invariant solutions and investigate problems of stabilizing periodic perturbations of the equilibrium position for a one-dimensional 6-velocity model. An exponentially fast stabilization of periodic perturbations of the equilibrium position to a traveling wave is established (with general periodic initial perturbations).

For a boundary value problem in the gradient theory of elasticity for the case of antiplane shear, we study invariant integrals independent of the integration contour. Using the method of differentiating the energy functionals with respect to domain shape, formulas for the energy derivative under domain perturbation are obtained. Conditions on the perturbation field are found that allow the derivative to be expressed as an integral over an arbitrary contour surrounding the crack. We construct invariant integrals corresponding to the crack rotation, translation, and quasi-static growth. For a straight crack, an analog of the Cherepanov–Rice integral is obtained, whose value determines the energy release rate during crack propagation.

The paper analyzes a complex heat transfer model in a bounded threedimensional domain, including a P1-approximation for the radiative transfer equation. The complex heat transfer process is modeled by a system consisting of a nonlinear parabolic equation for the temperature field and an elliptic equation for the radiative intensity, averaged over all directions. A statement of the initial-boundary value problem is presented in which the boundary values of the temperature and its normal derivative are known, while boundary conditions for the radiative intensity are not specified. Boundary conditions of this type for a diffusion model of complex heat transfer arise in the following situation: In order to specify a standard boundary condition for the radiative intensity, it is necessary to find a function describing the boundary blackness. If this function is unknown, it is natural to specify the heat fluxes at the boundary together with the boundary temperature instead of the boundary condition for radiative intensity. In this paper, we prove the asymptotic exponential stability of constant steady states without restrictions on the smallness of initial perturbations.