Robustness of Exponential Dichotomies in Infinite-Dimensional Dynamical Systems |
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Authors: | Victor A Pliss George R Sell |
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Institution: | (1) Faculty of Mathematics and Mechanics, St. Petersburg University, St. Petersburg, Russia;(2) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455 |
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Abstract: | In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation t
+A=B(t) , where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t)=B(, t) depends continuously on a parameter and there is an exponential dichotomy, or exponential trichotomy, at a value 0, then there is an exponential dichotomy, or exponential trichotomy, for all near 0.We present several illustrations indicating the significance of this robustness property. |
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Keywords: | Exponential dichotomy exponential trichotomy linear evolutionary equations ordinary differential equations Navier– Stokes equations nonlinear wave equation normal hyperbolicity partial differential equations robustness time-varying coefficients |
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