Unique determination of a time-dependent potential for wave equations from partial data

We consider the inverse problem of determining a time-dependent potential q, appearing in the wave equation ∂t2u−Δxu+q(t,x)u=0 in Q=(0,T)×Ω with T>0 and Ω a C2 bounded domain of Rn, n⩾2, from partial observations of the solutions on ∂Q. More precisely, we look for observations on ∂Q that allows to recover uniquely a general time-dependent potential q without involving an important set of data. We prove global unique determination of q∈L∞(Q) from partial observations on ∂Q. Besides being nonlinear, this problem is related to the inverse problem of determining a semilinear term appearing in a nonlinear hyperbolic equation from boundary measurements.