Well-posedness and ill-posedness of boundary-value problems for one class of fourth-order differential equations of Sobolev type

This paper is devoted to the study of the well-posedness of boundary-value problems for Sobolev-type differential equations
\begin{equation*} \frac{\partial^2}{\partial t^2}(Au)+Bu+h(x,y,t)Cu=f(x,y,t), \end{equation*}
in which $x$ is a point from the bounded domain $\Omega$ of the space $\mathbb{R}^n_x$, $y$ is a point from the bounded domain $G$ of the space $\mathbb{R}^m_y$, $t$ is a point of the interval $(0,T)$, $A$ and $B$ are second-order elliptic operators acting on variables $x_1,\ldots,x_n$, $C$ is a second-order elliptic operator acting on $y_1,\ldots,y_m$, and $h(x,y,t)$ and $f(x,y,t)$ are given functions. For these equations, we study the well-posedness in the S. L. Sobolev spaces of the initial-boundary-value and Dirichlet problems.