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

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.