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A hyperbolic type integro-differential equation with two weakly singularkernels is considered together with mixed homogeneous Dirichlet andnon-homogeneous Neumann boundary conditions. Existence and uniqueness of thesolution is proved by means of Galerkin's method. Regularity estimates areproved and the limitations of the regularity are discussed. The approachpresented here is also used to prove regularity of any order for models withsmooth kernels, that arise in the theory of linear viscoelasticity, under theappropriate assumptions on data.

Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity