An one-dimensional inverse problem for the wave equation

For the wave equation with inhomogeneity σ(x)um t + q(x)up a forward and an one-dimensional inverse problems are studied. Here m > 1 and p > 1 are real numbers. The forward problem is considered in the domain x > 0,t > 0 with zero initial data and Dirichlet boundary condition at x = 0. An unique solvability theorem of this problem is proved. The inverse problem is devoted to determining the coefficients σ(x) and q(x). As an additional information for recovering this coefficients, two forward problems with different Dirichlet data are considered and traces of the derivative of their solutions with respect to x are given at x =0 on a finite interval. For the inverse problem a local existence and uniqueness theorem is established.