This article is cited in 19 papers

Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect

V. V. Vlasov^a, J. Wu^b, G. R. Kabirova<sup>a

a</sup> M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia, Moscow
^b York University, Toronto, Canada

Abstract: We study functional differential equations with unbounded operator coefficients in Hilbert spaces such that the principal part of the equation is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. The well-posed solvability of initial boundary-value problems for the specified problems in weighted Sobolev spaces on the positive semi-axis is established. Our concern is spectra of operator-valued functions that are symbols of the specified equations in the autonomous case. In particular, the spectra of the Gurtin–Pipkin equation is studied, which is an integrodifferential equation modelling the heat propagation in media with memory.

UDC: 517.951

 English version: Journal of Mathematical Sciences, 2010, 170:3, 388–404

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