MiS Preprint Repository

Consider an abstract evolution problem in a Hilbert space $H$ $$\dot{u}=A(t)u+G(t,u)+f(t),u(0)=u_0,$$ where $A(t)$ is a linear, closed, densely defined operator in $H$ with domain independent of $t\ge 0$, $G(t,u)$ is a nonlinear operator such that $||G(t,u)||\le a(t)||u|{|}^p$, $p=const>1$, $||f(t)||\le b(t)$. We allow the spectrum of $A(t)$ to be in the right half-plane $\mathrm{Re}(\lambda )<{\lambda }_0(t)$, ${\lambda }_0(t)>0$, but assume that $li{m}_{t\to \mathrm{\infty }}{\lambda }_0(t)=0$.\ Under suitable assumption on $a(t)$ and $b(t)$ we prove boundedness of $||u(t)||$ as $t\to \mathrm{\infty }$. If $f(t)=0$, the Lyapunov stability of the zero solution to problem (1) with $u_0=0$ is established. For $f\ne 0$, sufficient conditions for Lyapunov stability are given. The novel point in the paper is the possibility for the linear operator $A(t)$ to have spectrum in the half-plane $\mathrm{\Re }(\lambda )<{\lambda }_0(t)$ with ${\lambda }_0(t)>0$ and $li{m}_{t\to \mathrm{\infty }}{\lambda }_0(t)=0$ at a suitable rate.