An abstract Dirichlet problem in the Hilbert space

In the present paper, we consider an abstract partial differential equation of the form $\frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{{\partial ^2 u}}{{\partial x^2 }} + A\left( {x,t} \right)u = f\left( {x,t} \right)$ , where $\left\{ {A\left( {x,t} \right):\left( {x,t} \right) \in \bar G} \right\}$ is a family of linear closed operators and $\bar G = G \cup \partial G,G$ is a suitable bounded region in the (x, t)-plane with boundary?G. It is assumed thatu is given on the boundary?G. The objective of this paper is to study the considered Dirichlet problem for a wide class of operatorsA(x, t). A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considered.