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

Zh. D. Totieva

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 2, pp. 58-66

Résumé

We consider the problem of determining the matrix kernel $K(t)=\mathrm{diag}(K_1, K_2, K_3)(t)$, $ 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 $u(x,t)=(u_1,u_2,u_3)(x,t),$ $x=(x_1,x_2,x_3) \in R^3,$ $x_3>0$. It is assumed that the coefficients of the system (density and elastic modulus) depend only on the spatial variable $x_3>0$. The source of perturbation of elastic waves is concentrated on the boundary of $x_3=0$ 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 $K_i(t)$, $ i=1,2,3$. As an additional condition, the value of the Fourier transform in $x_2$ of the function $u(x,t)$ is given on the surface $x_3=0$. 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.