Stability results of some abstract evolution equations

Abstract

The stability of the solution to the equation $\dot{u}=A(t)u+G(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. 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 $\Vert G(t,u)\Vert \le \alpha (t)\Vert u{\Vert }^p$, $p>1$, and the spectrum of $A(t)$ lies in the half-plane $\Real λ\le γ(t)$ 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 ${\int }_0^t\gamma (\xi )d\xi \to -\infty$ as $t\to \infty$, 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 ${\lim }_{t\to \infty }u(t)=0$ are proposed and justified.