Stability estimates for an inverse problem related to viscoelastic media

We consider a second-order hyperbolic integro-differential equation governing the third component of the displacement vector in viscoelasticity when the functions accounting for the properties of the viscoelastic medium and the external force are independent of coordinate x₃. We consider the inverse problem consisting in recovering two functions characterizing both the elastic and the viscoelastic moduli of the medium. As additional information to solve our inverse problem we use three measurements related to the solutions to three direct Cauchy problems with an impulse force supported on suitable planes. Prescribing Cauchy data on the lateral surface of the body assumed to occupy initially a cylindrical region, we obtain a conditional stability estimate for the solution to our inverse problem as well as a uniqueness theorem.