Well-posedness of Volterra integro-differential equations with singular kernels

N. A. Rautian

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 135-148.

Résumé

This paper is devoted to the study of integro-differential equations with unbounded operator coefficients in a Hilbert space. The principal part of the equations considered is an abstract hyperbolic equation perturbed by terms containing Volterra integral operators with singular kernels. Such integro-differential equations describe, for example, linear viscoelastic phenomena, heat propagation processes in media with memory (the Gurtin–Pipkin equation), averaging problems in perforated media (Darcy's law), etc. We establish conditions for the existence and uniqueness of strong and generalized solutions to initial-boundary-value problems for these equations in weighted Sobolev spaces on the positive semiaxis.

Détail

Keywords: integro-differential equation, operator function, well-posedness.

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