Quasi-periodic solutions of beam equations with the nonlinear terms depending on the time and space variables

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This article is devoted to the study of a beam equation with a constant potential and an $x$-periodic and $t$-quasiperiodic nonlinear term. It is proved that the equation admits small amplitude, linearly stable and $t$-quasiperiodic solutions under periodic boundary conditions for most values of the frequency vector and for most potentials. By utilizing the measure estimation of essentially finitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form. Based on an infinite dimensional KAM theorem, we prove the existence of quasi-periodic solutions.