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

Abstract

We consider the inverse problem of determining a time-dependent potential $q$, appearing in the wave equation ${\partial }_t^{2}u-{\mathrm{\Delta }}_{x}u+q(t,x)u=0$ in $Q=(0,T)\times \mathrm{\Omega }$ with $T>0$ and $\Omega$ a ${\mathcal{C}}^2$ bounded domain of ${\mathbb{R}}^n$, $n⩾2$, from partial observations of the solutions on $\partial Q$. More precisely, we look for observations on $\partial Q$ that allows to recover uniquely a general time-dependent potential $q$ without involving an important set of data. We prove global unique determination of $q\in L^{\infty }(Q)$ from partial observations on $\partial Q$. 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.