Boundary value problems for Sobolev type equations with irreversible operator coefficient of the highest derivatives

This paper is devoted to the study of the solvability of boundary-value problems for differential equations of the form
$$ (\alpha_0(t)+\alpha_1(t)\Delta)u_{tt}-Bu_t-Cu=f(x,t), $$
where $\Delta$ is the Laplace operator acting with respect to spatial variables and $B$ and $C$ are also second-order differential acting with respect to spatial variables. A feature of the equations considered is the condition that the functions $\alpha_0(t)$ and $\alpha_1(t)$ may not possess the fixed sign property on the range $(0,T)$ of the temporal variable; in particular, the operator $\alpha_0(t)+\alpha_1(t)\Delta$ may be irreversible at any point of the interval $(0,T)$, including any strictly inner segments. For problems considered, we prove theorems on the existence and uniqueness of regular solutions (i.e., solutions possessing all generalized derivatives in the Sobolev sense).