We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
56/2011
Stability result for abstract evolution problems
Alexander Ramm
Abstract
Consider an abstract evolution problem in a Hilbert space $H$ \begin{equation} \dot{u} = A(t)u+G(t,u)+f(t),\quad u(0)=u_0, \end{equation} where $A(t)$ is a linear, closed, densely defined operator in $H$ with domain independent of $t \geq 0$, $G(t,u)$ is a nonlinear operator such that $||G(t,u)|| \leq a(t)||u||^p$, $p=const>1$, $||f(t)||\le b(t)$. We allow the spectrum of $A(t)$ to be in the right half-plane $\mathop{\rm Re}(\lambda)<\lambda_0(t)$, $\lambda_0(t)>0$, but assume that $lim_{t \to \infty} \lambda_0(t)=0$.\\ Under suitable assumption on $a(t)$ and $b(t)$ we prove boundedness of $||u(t)||$ as $t \to \infty$. If $f(t)=0$, the Lyapunov stability of the zero solution to problem (1) with $u_0=0$ is established. For $f \neq 0$, 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 $\Re(\lambda)<\lambda_0(t)$ with $\lambda_0(t)>0$ and $lim_{t \to \infty} \lambda_0(t)=0$ at a suitable rate.
