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Almost periodicity of inhomogeneous parabolic evolution equations. (English) Zbl 1047.35078
Ruiz Goldstein, Gisèle (ed.) et al., Evolution equations. Proceedings of the conference, Blaubeuren, Germany, June 11–17, 2001 in honor of the 60th birthdays of Philippe Bénilan, Jerome A. Goldstein and Rainer Nagel. New York, NY: Marcel Dekker (ISBN 0-8247-0975-6/pbk). Lect. Notes Pure Appl. Math. 234, 299-318 (2003).

The authors deal with the problem of almost periodicity of solutions of the (parabolic) differential equation

u ' (t)=A(t)u(t)+f(t),(E)

on or on + . In the second case an initial condition of the form u(0)=xX= the underlying Banach space. Relying on such concepts like evolution family of bounded operators (on X), Yosida approximation, they prove existence of almost periodic solutions for the equation (E) on the real line , and the existence of asymptotically almost periodic solutions for the equation (E) on the half-line + . Conditions are formulated assuring exponential asymptotic stability for the corresponding homogeneous equation.

MSC:
35K90Abstract parabolic equations
34G10Linear ODE in abstract spaces
34C27Almost and pseudo-almost periodic solutions of ODE