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

• Well-posedness of a hyperbolic integro-differential equation is studied.
• The model arises in fractional viscoelasticity with two Mittag-Leffler type kernels.
• Homogeneous Dirichlet and nonhomogeneous Neumann boundary conditions are considered.
• Existence, uniqueness and regularity of the solution are proved by Galerkin's method.
• We extend the method to prove regularity of any order for models with smooth kernels.

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