Stability of solutions to abstract evolution equations with delay

Alexander Ramm

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Submission date: 03. Aug. 2011
Pages: 9
published in: Journal of mathematical analysis and applications, 396 (2012) 2, p. 523-527 
DOI number (of the published article): 10.1016/j.jmaa.2012.06.033
Bibtex
MSC-Numbers: 34E05, 35R30, 74J25, 34G20, 34K20, 37L05
Keywords and phrases: stability, evolution problems, abstract evolution equations, equations with delay
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Abstract:
An equation = A(t)u + B(t)F(t,u(t - τ)), u(t) = v(t),-τ ≤ t ≤ 0 is considered, A(t) and B(t) are linear operators in a Hilbert space H, = , F : H → H is a non-linear operator, τ > 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 ≥ 0, to be globally bounded, and to tend to zero as t →∞.