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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)].

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