Let $\Omega_T$
be some bounded simply connected region in $\mathbb{R}^2$
with $\partial\Omega_{T} = \bar{\Gamma}_{1}\cap\bar{\Gamma}_{2}$ . We seek a function $u(x,t),((x,t)\in\Omega_{T})$
with values in a Hilbert space $H$
which satisfies the equation $ALu(x,t) = Bu(x,t) + f(x,t,u,u_{t}),(x,t)\in\Omega_{T}$ , where $A(x,t),B(x,t)$
are families of linear operators
(possibly unbounded) with everywhere dense domain $D$
( $D$ does
not depend on $(x,t)$ ) in $H$ and
$Lu(x,t)= u_{tt}+ a_{11}u_{xx}+ a_{1}u_{t}+ a_{2}u_{x}$ . The values
$u(x,t);\partial u(x,t)/\partial n$
are given
in $\Gamma_1$ . This problem is not in general well posed in the sense of
Hadamard. We give theorems of uniqueness and stability of the
solution of the above problem.