Stability of solutions to abstract differential equations

Abstract

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, $\Vert F(t,u)\Vert \le c_0\Vert u{\Vert }^{1+p}$, $p=const>0$, $c_0=const>0$. It is not assumed that the spectrum $\sigma :=\sigma (A(t))$ of $A(t)$ lies in the fixed halfplane Re$z\le -\kappa$, 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.