The problem of determining the matrix kernel of the anisotropic viscoelasticity equations system

We consider the problem of determining the matrix kernel K(t)=diag(K1,K2,K3)(t), t>0, occurring in the system of integro-differential viscoelasticity equations for anisotropic medium. The direct initial boundary value problem is to determine the displacement vector function u(x,t)=(u1,u2,u3)(x,t), x=(x1,x2,x3)∈R3, x3>0. It is assumed that the coefficients of the system (density and elastic modulus) depend only on the spatial variable x3>0. The source of perturbation of elastic waves is concentrated on the boundary of x3=0 and represents the Dirac Delta function (Neumann boundary condition of a special kind). The inverse problem is reduced to the previously studied problems of determining scalar kernels Ki(t), i=1,2,3. As an additional condition, the value of the Fourier transform in x2 of the function u(x,t) is given on the surface x3=0. Theorems of global unique solvability and stability of the solution of the inverse problem are given. The idea of proving global solvability is to apply the contraction mapping principle to a system of nonlinear Volterra integral equations of the second kind in a weighted Banach space.