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

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Published: Jun 30, 2020
Keywords:
Second order evolution equation, partial differential equation, spectrum of a function, almost periodicity, asymptotic almost periodicity
Mathematics Subject Classification:
34G10, 35B15

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Nguyen Huu Tri
Division of Mathematics, Truong THPT Trung Van Phuong Trung Van, Nam Tu Liem, Hanoi, Vietnam
Bui Xuan Dieu
School of Applied Mathematics & Informatics Hanoi University of Science and Technology 01 Dai Co Viet road, Ha Noi, Vietnam
Vu Trong Luong
VNU University of Education, Vietnam National University Hanoi; 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
Nguyen Van Minh
Department of Mathematics and Statistics University of Arkansas at Little Rock, 2801 S University Ave Little Rock, AR 72204. USA

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 $f$.



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