Korean J. Math.  Vol 28, No 2 (2020)  pp.223-240
DOI: https://doi.org/10.11568/kjm.2020.28.2.223

Almost periodic solutions of periodic second order linear evolution equations

Nguyen Huu Tri, Bui Xuan Dieu, Vu Trong Luong, Nguyen Van Minh

Abstract


 The paper is concerned with periodic linear evolution equations of the form $x''(t)=A(t)x(t)+f(t)$, where $A(t)$ is a family of (unbounded) linear operators in a Banach space $X$, strongly and periodically depending on $t$, $f$ is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ${\mathbb R}$ as well as to have asymptotic almost periodic solutions on ${\mathbb R}^+$. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that  are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term $f$.


Keywords


Second order evolution equation, partial differential equation, spectrum of a function, almost periodicity, asymptotic almost periodicity

Subject classification

34G10, 35B15

Sponsor(s)



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