V. V. Vlasov

Spectral Problems Arising in the Theory of Differential Equations with Delay

In this article we present a review of results on asymptotic behavior and stability of strong solutions for functional differential equations (FDE). We also formulate several results about spectral properties (completeness and basisness) of exponential solutions of the above-mentioned equations. It is relevant to emphasize that our approach for the research of FDE is based on the spectral analysis of operator pencils that are symbols (characteristic quasi-polynomials) with operator coefficients. The article is divided into two parts. The first part is devoted to the research on FDE in a Hilbert space; the second part is devoted to the research on FDE in a finite-dimensional space.

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.

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.

Well-Posed Solvability of Functional-Differential Equations with Unbounded Operator Coefficients

We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.