Well-Posedness of Volterra Integro-Differential Equations with Fractional Exponential Kernels

Abstract

Well-defined solvability of initial boundary value problems for integro-differential equations with unbounded operator coefficients in Hilbert spaces is established in weighted Sobolev spaces on the positive semi-axis. The principal part of such equations is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. These equations can be regarded as an abstract generalization of the Gurtin-Pipkin integro-differential equation that describes heat transfer in materials with memory and has a number of other applications. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations.