An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic functions. In addition to constructing a bounded strong solution, we determine a time-dependent source coefficient via an integral overdetermination condition ensuring well-posedness. After reducing the inverse problem to a direct one, we first establish existence and uniqueness of solutions to an associated problem on finite time intervals. We then extend these solutions to half-lines and construct a bounded strong solution on the whole real line as a limit of such extensions, and subsequently establish its uniqueness. In particular, periodic and almost periodic data yield periodic and almost periodic solutions.