Boundary value problems for third-order pseudoelliptic equations with degeneration

We study the solvability in Sobolev spaces of the Dirichlet problem and other elliptic problems for the differential equations
\begin{equation} u_{tt}+\alpha(t)\frac{\partial}{\partial t}(\Delta u)+Bu=f(x,t)\tag{*} \end{equation}
$x\in\Omega\subset\mathbb{R}^n$, $t\in(0,T),$ where $\Delta$ if the Laplace operator acting in the variables $x_1,\dots, x_n$ and $B$ is a second-order elliptic operator acting in the same variables $x_1,\dots, x_n$. A feature of the equations ($\ast$) is that the sign of the function is not fixed in them. Existence and uniqueness theorems for regular solutions (having all generalized Sobolev's derivatives in the equation) are proved for the problems under study.