Free and forced vibrations of nonlinear wave equations in ball

Abstract

First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation ( ∂ t 2−△) u= f( r, u) in n-dimensional ball B a with center at the origin and radius a, where f is smooth, monotone decreasing in u, and satisfies f( r,0)=0. f( r, u) has asymptotic properties f(r,u)=O(u 3)(u→0 and u→±∞) . For n=1,3 we shall show the existence of infinitely many radially symmetric time-periodic solutions with different periods of irrational multiple of a. Second, we shall deal with BVP for a forced nonlinear wave equation ( ∂ t 2−△) u= εg( r, t, u), where g is T-periodic in t and ε is a small parameter. Under some Diophantine condition on a/ T we shall show the existence of time-periodic solutions of the BVP. For 1⩽ n⩽5 we shall construct infinitely many T satisfying the above Diophantine inequality, using asymptotic expansions of the zero points of the Bessel functions.