Stability in the determination of a time-dependent coefficient for wave equations from partial data

Let Ω be a C2 bounded domain of Rn, n⩾2, and fix Q=(0,T)×Ω with T>0. We consider the stability in the inverse problem of determining a time-dependent coefficient of order zero q, appearing in a Dirichlet initial-boundary value problem for a wave equation ∂t2u−Δxu+q(t,x)u=0 in Q, from partial observations on ∂Q. The observation is given by a boundary operator associated to the wave equation. Using suitable geometric optics solutions and Carleman estimates, we prove a stability estimate in the determination of q from the boundary operator.