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 ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+q(t,x)u=0$ in $Q=(0,T)\times \mathrm{\Omega }$ with $\mathrm{\Omega }$ a $C^2$ bounded domain of ${\mathbb{R}}^n$, $n\ge 2$, from partial observations on $\mathrm{\partial }Q$. 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.