Free and forced vibrations of nonlinear wave equations in ball
Free and forced vibrations of nonlinear wave equations in ball
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DOI：
10.1016/j.jde.2004.04.014
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发表时间：
2004-09
影响因子：
2.4
通讯作者：
M. Yamaguchi
中科院分区：
文献类型：
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作者：
M. Yamaguchi
首先，我们将讨论一个非线性径向对称波动方程（Δ t 2-△）u=f（r，u）在n维球Ba中的自由振动，中心在原点，半径为a，其中f是光滑的，关于u单调下降，满足f（r，0）=0。f（r，u）具有渐近性质f（r，u）=O（u ~ 3）（u→0，u→±∞）.对于n= 1，3，我们将证明存在无穷多个具有不同周期的a的无理倍数的径向对称时间周期解。第二，我们将讨论强迫非线性波动方程（ε t2-△）u=εg（r，t，u）的边值问题，其中g是关于t的T-周期，ε是一个小参数。在a/T上的某些丢番图条件下，我们将证明边值问题时间周期解的存在性。对于1 n 5，我们将构造满足上述丢番图不等式的无穷多个T，使用贝塞尔函数零点的渐近展开式。
First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation (∂t2−△)u=f(r,u) in n-dimensional ball Bawith 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(u3)(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 (∂t2−△)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.