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 $\|G(t,u)\|\le \alpha(t)\|u\|^p$, $p>1$, and the spectrum of $A(t)$ lies in the half-plane $\Real \lambda \le \gamma(t)$ 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 $\int_0^t \gamma(\xi)\, d\xi \to -\infty$ 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.