A Multidimensional Identification Problem Related to a Hyperbolic Integro-Differential Equation

We prove a global in time existence and uniqueness theorem for the identification of a relaxation kernel h entering a hyperbolic integro-differential equation, related to a convex cylinder with a smooth lateral surface, when the coefficient h is assumed to depend on time and one space variable and general additional conditions are provided. A continuous dependence result for the identification problem is also stated. Finally, a separate proof concerning the existence and uniqueness of the solution to the related direct integro-differential problem is also given in a suitable functional space. Moreover, the dependence of such a solution with respect to the relaxation kernel is fully analysed.