Weak solutions of hyperbolic even-order operator-differential equations with variable domains

We prove the existence and uniqueness of weak solutions $u(t)\in L_2(]0,T[,H)$ of boundary value problem for a two-term even-order hyperbolic operator-differential equation with unbounded operator coefficient $A(t)$, having $t$-depending domain $D(A(t))$. It is shown that for a smooth right-hand part the weak solutions of boundary value problem are smooth, i. e. they satisfy the equation almost everywhere on $]0,T[$ in $H$ and the boundary conditions in the usual sense. An example of the new correct boundary value problem for fourth-order partial differential equation with unsteady boundary conditions on the space variables is given.