Periodic Oscillations of a Linear Wave Equation with a Small Time-periodic Potential

We shall consider BVP to a 3-dimensional radially symmetric lin-ear wave equation with a small time-periodic potential $\partial_{t}^{2}u-\triangle u+rau+\epsilon f(x, \omega t)u=0, $ $(x, t)\in D\cross R_{t}^{1} $, where $D $ is the 3-ball, $f(x, \theta) $ is $2\pi$-periodic in $\theta $ and smooth in $(x, \theta) $, $ra $ is a positive constant, $\epsilon $ is a small parameter and $\omega $ is a positive constant depending on $\epsilon $. We shall show that BVP has families of periodic solutions with periods $2\pi’\omega(\epsilon) $ for $\epsilon\in\Lambda $ , where $\Lambda $ is contained in a neighborhood of $0 $ , and is uncountable and has Lebesgue measure zero. The solutions bifurcate from each normal mode of $\partial_{t}^{2}u-\triangle u+$ $rau=0 $.