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Bounded solutions of linear periodic abstract parabolic equations. (English) Zbl 0673.35041
The author studies the asymptotic behavior of the evolution operator G(t,s) associated with a family of generators of analytic semigroup $$\{A(t):\quad t\in {\mathbb{R}}\}$$ with periodic condition $$A(t+T)=A(t)$$ in some Banach space. Some new maximal regularity results for the bounded solution in unbounded time intervals were established. The results were based on a series estimates on the operator G(t,s). The convergences of the bounded solutions to the parabolic type equation has $$t\to +\infty$$ (or $$t\to -\infty)$$ in some suitable sense are the consequences of the regularity results. An example of a linear parabolic nonhomogeneous equation is presented to illustrate the main results.
Reviewer: J.Yong

##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 47D03 Groups and semigroups of linear operators 35B65 Smoothness and regularity of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs
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