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Robustness of exponential dichotomies in infinite-dimensional dynamical systems. (English) Zbl 0941.37052

Summary: The authors examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on a Banach space \(W\). These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation \(\partial_t v+Av= B(t)v\), where the linear operator \(B(t)\) is the evaluation of a suitable Fréchet derivative along a given solution in the set \(K\) in \(W\). The main objective is to show, under reasonable conditions, that if \(B(t)=B(\lambda,t)\) depends continuously on a parameter \(\lambda\in \Lambda\) and there is an exponential dichotomy, or exponential trichotomy, at a value \(\lambda_0\in\Lambda\), then there is an exponential dichotomy, or exponential trichotomy, for all \(\lambda\) near \(\lambda_0\). They present several illustrations indicating the significance of this robustness property.

MSC:

37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems
35Q30 Navier-Stokes equations
47H20 Semigroups of nonlinear operators
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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