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In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin-Voigt friction.

In this paper, we study the problem on small motions of an ideal relaxing fluid that fills a uniformly rotating or fixed container. We prove a theorem on uniform strong solvability of the corresponding initial-boundary value problem. In the case where the system does not rotate, we find an asymptotic behavior of the solution under the stress of special form. We investigate the spectral problem associated with the system under consideration. We obtain results on localization of the spectrum, on essential and discrete spectrum, and on spectral asymptotics. For nonrotating system in zero-gravity conditions we prove the multiple basis property of a special system of elements. In this case, we find an expansion of the solution of the evolution problem in the special system of elements.

In this paper, we investigate a problem on small motions of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. The initial boundary value problem is reduced to the Cauchy problem for a first-order differential operator equation on a Hilbert space. Properties of the resulting operator matrices, which are coefficients of the equation, are studied. Theorems on solvability of the Cauchy problem and the initial boundary value problem are proven.

In this paper, we consider the hyperbolic thermoelastic system of memory type and the corresponding system of abstract integro-differential equations. We study forced motions of this system of abstract integro-differential equations and investigate the asymptotic behavior of solutions to this system as time goes to infinity.

A model of viscoelastic barotropic Maxwell fluid is investigated. The unique solvability theorem is proved for the corresponding initial-boundary value problem. The associated spectral problem is studied. We prove statements on localization of the spectrum, on the essential and discrete spectra, and on asymptotics of the spectrum.

A model of a viscoelastic compressible Maxwell fluid is studied. This model is described by a system of partial integro-differential equations with appropriate boundary and initial conditions. An abstract analog of the problem under study is also considered. It is proved that a uniformly exponentially stable C0-semigroup emerges in this problem. Based on this fact, an estimate for the solution of the evolution problem is derived in the case where the external load is close to being almost periodic.

In this paper, mathematical models of compressible viscoelastic Maxwell, Oldroyd, and Kelvin–Voigt fluids are derived. A model of rotating viscoelastic barotropic Oldroyd fluid is studied. A theorem on strong unique solvability of the corresponding initial-boundary value problem is proved. The spectral problem associated with such a system is studied. Results on the spectrum localization, essential and discrete spectra, and spectrum asymptotics are obtained. In the case where the system is in the weightlessness state and does not rotate, results on multiple completeness and basis property of a special system of elements are proved. In such a case, under the assumption the viscosity is sufficiently large, an expansion of the solution of the evolution problem with respect to a special system of elements is obtained.

An eigenvalue problem associated with small movements of a viscoelastic body fixed on the boundary of a bounded domain is studied. The spectrum of the problem is proved to lie in a vertical strip bounded away from the imaginary axis and to be symmetric about the real axis. The essential spectrum of the problem consists of a finite number of points on the real axis. There are two sequences of complex conjugate eigenvalues condensing toward infinity. Under certain additional conditions, the spectrum that does not lie on the real axis is bounded away from it.

The uniform exponential stability of a C0-semigroup with generator of a special form is proved. Such semigroups arise in the study of various problems of the theory of viscoelasticity. The proved statement is applied to the study of the asymptotic behavior of solutions in the problem of small motions of a viscoelastic body subject to driving forces of a special form.

The problem of small movements of a viscoelastic body of parabolic type is studied in the paper. The unique strong solvability of the corresponding initial-boundary value problem is proved. The spectrum and the properties of root elements of the emerging operator block are studied. More precisely, the theorem about both the essential and the discrete spectrum of the main operator block is proved. The asymptotic formula for the series of eigenvalues condensing at infinity is found. Completeness and the basis property of the system of root elements of the main operator are established. Presentations for a solution of the original second-order integrodifferential equation are found both in the form of contour integrals and expansions in the system of eigenvectors of some operator pencil. A certain statement concerning stabilization of solutions to the evolution problem is proved. In the last section, the case of a synchronously isotropic medium of parabolic type is studied as a particular case of the model considered.

We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.

We study an evolution problem on small motions of the ideal rotating relaxing fluid in bounded domains. We begin from the problem posing. Then we reduce the problem to a second-order integrodifferential equation in a Hilbert space. Using this equation, we prove a strong unique solvability problem for the corresponding initial-boundary value problem.

The spectral problem associated with the normal oscillations of a rotating “viscous fluid + system of ideal fluids” hydrodynamic system is examined. Asymptotic formulas are derived for part of the branches of the eigenmodes and observations are made on the limiting spectrum for the problem.