  We study the problem of determining uniquely a time-dependent singular
potential $q$, appearing in the wave equation $\partial_t^2u-\Delta_x
u+q(t,x)u=0$ in $Q=(0,T)\times\Omega$ with $T>0$ and $\Omega$ a $ \mathcal C^2$
bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique
determination of some singular time-dependent coefficients from observations on
$\partial Q$. Then, by weakening the singularities of the set of admissible
coefficients, we manage to reduce the set of data that still guaranties unique
recovery of such a coefficient. To our best knowledge, this paper is the first
claiming unique determination of unbounded time-dependent coefficients, which
is motivated by the problem of determining general nonlinear terms appearing in
nonlinear wave equations.
