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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 \[ u'(t) = A(t)u(t) + f(t),\tag{E} \] on \(\mathbb{R}\) or on \(\mathbb{R}_+\). In the second case an initial condition of the form \(u(0)= x\in X=\) 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 \(\mathbb{R}\), and the existence of asymptotically almost periodic solutions for the equation (E) on the half-line \(\mathbb{R}_+\). Conditions are formulated assuring exponential asymptotic stability for the corresponding homogeneous equation.
For the entire collection see [Zbl 1027.00022].

35K90 Abstract parabolic equations
34G10 Linear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations