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Bounded solutions for abstract time-periodic parabolic equations with nonconstant domains. (English) Zbl 0729.34041
Let E be a Banach space and $$A(t)$$, $$t\in {\mathbb{R}}$$ a family of closed linear operators with time dependent and nondense domains which generate analytic semigroups (not continuous at $$t=0)$$. The author studies the problem $$u'(t)=A(t)u(t)+f(t)$$, $$t\in I$$; $$u(t_ 0)=x$$ in an unbounded interval $$I\subseteq {\mathbb{R}}$$ under the periodicity condition $$A(t+T)=A(t)$$. If u(t,s) is the fundamental solution of this problem and if in the unit circle there are no elements of the spectrum of $$U(t+T,t)$$, $$t\in {\mathbb{R}}$$ then the author proves the existence and representation formulas of bounded and periodic solutions. The same problem has been treated in the constant domain case by A. Lunardi [Proc. R. Soc. Edinb., Sect. A 110, No.1/2, 135-159 (1988; Zbl 0673.35041)].

##### MSC:
 34G10 Linear differential equations in abstract spaces 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations