Preprint 56/2011

Stability result for abstract evolution problems

Alexander Ramm

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Submission date: 29. Aug. 2011
Pages: 9
MSC-Numbers: 34E05, 35R30, 74J25
Keywords and phrases: stability, evolution problems
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Consider an abstract evolution problem in a Hilbert space H

˙u = A(t)u + G(t,u )+ f(t),  u(0) = u0,

where A(t) is a linear, closed, densely defined operator in H with domain independent of t 0, G(t,u) is a nonlinear operator such that ||G(t,u)||≤ a(t)||u||p, p = const > 1, ||f(t)||≤ b(t). We allow the spectrum of A(t) to be in the right half-plane Re(λ) < λ0(t), λ0(t) > 0, but assume that limt→∞λ0(t) = 0.
Under suitable assumption on a(t) and b(t) we prove boundedness of ||u(t)|| as t →∞. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u0 = 0 is established. For f0, 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 (λ) < λ0(t) with λ0(t) > 0 and limt→∞λ0(t) = 0 at a suitable rate.

03.04.2017, 12:08