N. A. Rautian

Spectral analysis of hyperbolic Volterra integro-differential equations

Volterra integro-differential equations with unbounded operator coefficients in a Hilbert space are studied. The equations under consideration are abstract hyperbolic equations perturbed by terms containing Volterra integral operators. These equations can be realized as integro-differential partial differential equations arising in the theory of viscoelasticity (see [4, 6]) and as the Gurtin–Pipkin integro-differential equations (see [1, 6]), which describe the process of heat propagation in media with memory at finite rate; these equations arise also in problems of averaging in multiphase media (Darcy’s law) (see [9]).

Representation of Solutions of Integro-Differential Equations with Kernels Depending on the Parameter

Integro-differential equations with unbounded operator coefficients in a separable Hilbert space are studied. These equations are an abstract form of the Gurtin–Pipkin-type equation, which describes finite-speed propagation of heat in media with memory. A representation of strong solutions of these equations is derived in the form of the sums of series in exponents that correspond to the spectral points of operator-functions that are the symbols of these equations.

Study of Volterra integro-differential equations arising in viscoelasticity theory

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.

On the asymptotic behavior of solutions of integro-differential equations in a Hilbert space

We consider integro-differential equations that are an abstract form of the well-known Gurtin-Pipkin equation. We obtain representations of strong solutions of these equations in the form of series in the exponentials corresponding to points of spectrum of the symbols of such equations.

Study of Volterra Integro-Differential Equations with Kernels Depending on a Parameter

We carry out spectral analysis of operator functions that are the symbols of integro-differential equations with unbounded operator coefficients in a separable Hilbert space. The structure and localization of the spectrum of operator functions which are symbols of these equations play an important role in studies of the asymptotic behavior of their solutions.

Study of functional-differential equations with unbounded operator coefficients

Functional-differential and integro-differential equations with the principal part being an abstract hyperbolic equation perturbed by terms with unbounded variable operator coefficients multiplying variable delays are studied. Additionally, Volterra integral operators are considered. For the equations under study, the well-posedness of initial value problems in Sobolev spaces of vector functions is proved. In the autonomous case, spectral analysis of the operator functions that are the symbols of the indicated equations is performed.

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.

Well-Posedness and Spectral Analysis of Hyperbolic Volterra Equations of Convolution Type

We study the correct solvability of abstract integrodifferential equations in Hilbert space generalizing integrodifferential equations arising in the theory of viscoelasticity. The equations under considerations are the abstract hyperbolic equations perturbed by the terms containing Volterra integral operators. We establish the correct solvability in the weighted Sobolev spaces of vector-valued functions on the positive semiaxis. We also provide the spectral analysis of operator-valued functions which are the symbols of these equations.

Integrodifferential equations in viscoelasticity theory

We prove the correct solvability of the initial problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Such equations occur in many problems of the theory of viscoelasticity with memory and the heat transfer theory.