Stability result for abstract evolution problems

Alexander Ramm

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Submission date: 29. Aug. 2011
Pages: 9
published in: Mathematical methods in the applied sciences, 36 (2013) 4, p. 422-426 
DOI number (of the published article): 10.1002/mma.2603
Bibtex
with the following different title: A stability result for abstract evolution problems
MSC-Numbers: 34E05, 35R30, 74J25
Keywords and phrases: stability, evolution problems
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Abstract:
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, 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 lim_t→∞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 f≠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 ℜ(λ) < λ₀(t) with λ₀(t) > 0 and lim_t→∞λ₀(t) = 0 at a suitable rate.