Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data

Abstract

We consider the inverse problem of determining a time-dependent damping coefficient $a$ and a time-dependent potential $q$, appearing in the wave equation ${\partial }_t^{2}u-{\Delta }_{x}u+a(t,x){\partial }_{t}u+q(t,x)u=0$ in $Q=(0,T)\times \Omega$, with $T>0$ and $\Omega$ a ${\mathcal{C}}^2$ bounded domain of ${\mathbb{R}}^n$, $n\ge 2$, from partial observations of the solutions on $\partial Q$. More precisely, we look for observations on $\partial Q$ that allow to determine uniquely a large class of time-dependent damping coefficients $a$ and time-dependent potentials $q$ without involving an important set of data. We prove global unique determination of $a\in W^{1,p}(Q)$, with $p>n+1$, and $q\in L^{\infty }(Q)$ from partial observations on $\partial Q$.