Stability results of some abstract evolution equations

The stability of the solution to the equation $\dot{u} = A(t)u + G(t,u)+f(t)$,$t\ge 0$,$u(0)=u_0$isstudied\mathrm{.}Here$A(t)$ is a linear operator in a Hilbert space $H$ and $G(t,u)$ is a nonlinear operator in $H$ for any fixed $t\ge 0$. We assume that $\mathrm{\parallel }G(t,u)\mathrm{\parallel }\le \alpha (t)\mathrm{\parallel }u{\mathrm{\parallel }}^p$, $p>1$, and the spectrum of $A(t)$ lies in the half-plane $\text{Real}\lambda \le \gamma (t)$ where $\gamma (t)$ can take positive and negative values. We proved that the equilibrium solution $u=0$ to the equation is Lyapunov stable under persistantly acting perturbations $f(t)$ if $\sup_{t\ge 0}\int_0^t \gamma(\xi), d\xi <\infty$and$\int_0^\infty \alpha(\xi), d\xi<\infty$. In addition, if ${\int }_0^t\gamma (\xi )d\xi \to -\mathrm{\infty }$ as $t\to \mathrm{\infty }$, then we proved that the equilibrium solution $u=0$ is asymptotically stable under persistantly acting perturbations $f(t)$. Sufficient conditions for the solution $u(t)$ to be bounded and for ${\lim }_{t\to \mathrm{\infty }}u(t)=0$ are proposed and justified.