A global solvability of a two‐dimensional kernel determination problem for a viscoelasticity equation

Abstract

Under study is the two‐dimensional inverse problem of determining the convolutional kernel of the integral term in a viscoelasticity equation. The direct problem is represented by a generalized initial‐boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta function. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions twice continuously differentiable in the time variable and analytic in the space variable. We apply the methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions.