Geodesic
Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 44-59

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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. 
@article{CMFD_2010_35_a3,
     author = {V. V. Vlasov and J. Wu and G. R. Kabirova},
     title = {Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {44--59},
     publisher = {mathdoc},
     volume = {35},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2010_35_a3/}
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V. V. Vlasov; J. Wu; G. R. Kabirova. Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 44-59. http://geodesic.mathdoc.fr/item/CMFD_2010_35_a3/