Inverse problems of finding the lowest coefficient in the elliptic equation

The article is devoted to the study of problems of finding the non-negative coefficient $q(t)$ in the elliptic equation
$$u_{tt}+a^2\Delta u-q(t)u=f(x,t)$$
($x=(x_1,\ldots,x_n)\in\Omega\subset \mathbb{R}^n$, $t\in (0,T)$, $0<T<+\infty$, $\Delta$ — operator Laplace on $x_1, \ldots, x_n$). These problems contain the usual boundary conditions and additional condition ( spatial integral overdetermination condition or boundary integral overdetermination condition). The theorems of existence and uniqueness are proved.