Duhamel's method in inverse problems for the wave equation. I

This paper is devoted to inverse problems of recovering the time dependent source and coefficients of the wave equation. The mixed problem with the Neumann boundary condition is considered. A certain weighted boundary integral with solution in question is used as overdetermination. To determine unknown source we use Duhamel’s method and get a Volterra type equation of the first and second kind. The kernel of this equation depends depends on the solution to an auxiliary mixed problem and the secodn order derivatives of the solution. A local boundary straightening and a special change of variables are used to get necessary estimates. To determine unknown time dependent coefficients we use the successive approximation method and contraction mapping principle.