On solvability of nonlocal boundary value problem for a differential equation of composite type

We study the solvability in anisotropic Sobolev spaces of nonlocal in time problems for the differential equations of composite (Sobolev) type
$$u_{tt}+\left(\alpha\frac{\partial}{\partial t}+\beta\right)\Delta u+\gamma u=f(x,t),$$
$x = (x_1,\ldots , x_n) \in\Omega\subset R^n$, $t\in(0, T),$ $0 < T < +\infty$, $\alpha, \beta,$ and $\gamma$ are real numbers, and $f(x, t)$ is a given function. We prove theorems of existence and non-existence, uniqueness and non-uniqueness for regular solutions, those having all generalized Sobolev derivatives in the equation.