A Characteristic Equation For Non-Autonomous Partial Functional Differential Equations (2000)
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BibTeX
@MISC{Gühring00acharacteristic,
author = {Gabriele Gühring and Frank Räbiger and Roland Schnaubelt},
title = {A Characteristic Equation For Non-Autonomous Partial Functional Differential Equations},
year = {2000}
}
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Abstract
. We characterize the exponential dichotomy of non--autonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case. From this we deduce robustness results. We further study the almost periodicity of solutions to the inhomogeneous equation. Our approach is based on the spectral theory of evolution semigroups. 1. Introduction For the autonomous partial functional differential equation u(t) = Au(t) + Lu t ; t 0; u(t) = OE(t); \Gammar t 0; (1.1) there is a well developed semigroup approach; in particular, a powerful spectral theory is available. Here we assume that A generates a C 0 --semigroup V (\Delta) on a Banach space X. Further, r 0, OE 2 E := C([\Gammar; 0]; X); L 2 L(E; X), and we let u t () := u(t + ) for 2 [\Gammar; 0], t 0, and u : [\Gammar; 1) ! X. Then the operator AL := d d ; D(AL ) := fOE 2 C 1 ([\Gammar; 0]; X) : OE(0) 2 D(A); OE 0 (0) = AOE(0) + ...
