Quasi-periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables. (December 2019)

• Record Type: Journal Article
• Authors: WANG, Yi
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 37K55 -- 70K43 -- 70K45 -- 70K40; Completely resonant beam equation -- Quasi-periodically forced -- x-dependent -- Quasi-periodic solution -- Normal form; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111585
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This article is devoted to the study of a completely resonant beam equation with an x -periodic and t -quasi-periodic nonlinear term. It is proved that the equation admits small amplitude, linearly stable and quasi-periodic solutions for most values of the frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form. We show an infinite dimensional KAM theorem for non-autonomous beam equations, which is applied to prove the existence of quasi-periodic solutions.