  The global existence and stability of the solution to the delay differential
equation (*)$\dot{u} = A(t)u + G(t,u(t-\tau)) + f(t)$, $t\ge 0$, $u(t) = v(t)$,
$-\tau \le t\le 0$, are studied. Here $A(t):\mathcal{H}\to \mathcal{H}$ is a
closed, densely defined, linear operator in a Hilbert space $\mathcal{H}$ and
$G(t,u)$ is a nonlinear operator in $\mathcal{H}$ continuous with respect to
$u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $\Re
\lambda \le \gamma(t)$, where $\gamma(t)$ is not necessarily negative and
$\|G(t,u)\| \le \alpha(t)\|u\|^p$, $p>1$, $t\ge 0$. Sufficient conditions for
the solution to the equation to exist globally, to be bounded and to converge
to zero as $t$ tends to $\infty$, under the non-classical assumption that
$\gamma(t)$ can take positive values, are proposed and justified.
