Stability of solutions to some abstract evolution equations with delay

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