  Large time behavior of solutions to abstract differential equations is
studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t),
\quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$,
$u=u(t)\in H$, $H$ is a Hilbert space, $t\in \R_+:=[0,\infty)$, $A(t)$ is a
linear dissipative operator: Re$(A(t)u,u)\le -\gamma(t)(u,u)$, %$\gamma(t)\ge
0$, $F(t,u)$ is a nonlinear operator, $|F(t,u)|\le c_0|u|^p$, $p>1$, $c_0,p$
are positive constants, $|b(t)|\le \beta(t),$ $\beta(t)\ge 0$ is a continuous
function.
  Sufficient conditions are given for the solution $u(t)$ to problem (*) to
exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on
$|u(t)|$ is given. This bound implies the relation $\lim_{t\to \infty}|u(t)|=0$
under suitable conditions on $\gamma(t)$ and $\beta(t)$.
