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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
50/2011

Stability of solutions to abstract evolution equations with delay

Alexander Ramm

Abstract

An equation $\dot{u}=A(t)u +B(t)F(t,u(t-\tau))$, $u(t)=v(t), -\tau \leq t \leq 0$ is considered, $A(t)$ and $B(t)$ are linear operators in a Hilbert space $H$, $\dot{u}=\frac{du}{dt}$, $F: H \to H$ is a non-linear operator, $\tau > 0$ is a constant. Under some assumption on $A(t), B(t)$ and $F(t,u)$ sufficient condittions are given for the solution $u(t)$ to exist globally, i.e, for all $t \geq 0$, to be globally bounded, and to tend to zero as $t \to \infty$.

Received:
Aug 3, 2011
Published:
Aug 8, 2011
MSC Codes:
34E05, 35R30, 74J25, 34G20, 34K20, 37L05
Keywords:
stability, evolution problems, abstract evolution equations, equations with delay

Related publications

inJournal
2012 Repository Open Access
Alexander G. Ramm

Stability of solutions to abstract evolution equations with delay

In: Journal of mathematical analysis and applications, 396 (2012) 2, pp. 523-527