A stability result for abstract evolutionproblems

Consider an abstract evolution problem in a Hilbert space H where A(t) is a linear, closed, densely defined operator in H with domain independent of t≥0 and G(t,u) is a nonlinear operator such that G(t,u)a(t) up, p=const>1, f(t)≤b(t). We allow the spectrum of A(t) to be in the right half-plane Re...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2013-03, Vol.36 (4), p.422-426
Main Author: Ramm, A. G
Format: Article
Language:English
Subjects:
Evolution
Inequalities
Mathematical analysis
Nonlinearity
Operators
Stability
Studies
Online Access:Get full text
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Summary:Consider an abstract evolution problem in a Hilbert space H where A(t) is a linear, closed, densely defined operator in H with domain independent of t≥0 and G(t,u) is a nonlinear operator such that G(t,u)a(t) up, p=const>1, f(t)≤b(t). We allow the spectrum of A(t) to be in the right half-plane Re(λ)0, but assume that limt[arrow right][infin]λ0(t)=0. Under suitable assumptions on a(t) and b(t), the boundedness of u(t) as t[arrow right][infin] is proved. If f(t)=0, the Lyapunov stability of the zero solution to problem (1) with u0=0 is established. For f≠0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half-plane Re(λ)0 and limt[arrow right][infin]λ0(t)=0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT]
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.2603