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

We consider the stability in the inverse problem consisting of the determination of a time-dependent coefficient of order zero $q$, appearing in a Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(t,x)u=0$ in $Q=(0,T)\times\Omega$ with $\Omega$ a $C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations on $\partial Q$. The observation is given by a boundary operator associated to the wave equation. Using suitable complex geometric optics solutions and Carleman estimates, we prove a stability estimate in the determination of $q$ from the boundary operator.