On the solvability of the inverse problems of parameter recovery in elliptic equations

We study solvability of the inverse problems of finding, alongside the solution $u(x,t)$, the positive parameter $\alpha$ in the differential equations
$$ u_{tt}+\alpha\Delta u-\beta u=f(x,t),\quad\alpha u_{tt}+\Delta u-\beta u=f(x,t), $$
where $t\in(0,T)$, $x=(x_1,\dots,x_n)\in\Omega\subset\mathbb{R}^n$, and $\Delta$ – the Laplace operator in variables $x_1,\dots,x_n$. As a complement to the boundary conditions defining a well-posed boundary value problem for elliptic equations, we use the conditions of the linear final integral overdetermination. We prove the existence and uniqueness theorems for regular solutions, those having all generalized in the S. L. Sobolev sense derivatives in the equation.