Large-time behavior of solutions to evolution problems

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