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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.

Received:
Aug 29, 2011
Published:
Sep 21, 2011
MSC Codes:
34E05, 35R30, 74J25
Keywords:
stability, evolution problems

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