An abstract Dirichlet problem in the Hilbert space

In the present paper, we consider an abstract partial differential equation of the form $frac{{partial ^2 u}}{{partial t^2 }} - frac{{partial ^2 u}}{{partial x^2 }} + Aleft( {x,t} right)u = fleft( {x,t} right)$ , where $left{ {Aleft( {x,t} right):left( {x,t} right) in bar G} right}$ 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.