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