An inverse problem for the wave equation with nonlinear dumping

We study the inverse problem of recovering a coefficient at the nonlinearity in a second order hyperbolic equation with nonlinear damping. The unknown coefficient depends on one space variable $x$. Also, we consider the process of wave propagation along the semiaxis $x>0$ given the derivative with respect to $x$ at $x=0$. As additional information in the inverse problem we consider the trace of a solution to the initial boundary value problem on a finite segment of the axis $x=0$ and find the conditions for unique solvability of the direct problem. We also establish a local existence theorem and a global stability estimate for a solution to the inverse problem.