Stability of solutions to abstract differential equations

A sufficient condition for asymptotic stability of the zero solution to an abstract nonlinear evolution problem is given. The governing equation is $\dot{u}=A(t)u+F(t,u),$ where $A(t)$ is a bounded linear operator in Hilbert space $H$ and $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^{1+p}$, $p=const >0$, $c_0=const>0$. It is not assumed that the spectrum $\sigma:=\sigma(A(t))$ of $A(t)$ lies in the fixed halfplane Re$z\leq -\kappa$, where $\kappa>0$ does not depend on $t$. As $t\to \infty$ the spectrum of $A(t)$ is allowed to tend to the imaginary axis.