高维波动方程的Cauchy问题存在时间周期解的充要条件

A Necessary and Sufficient Condition on the Existence of Time Periodic Solutions to the Cauchy Problemof Wave Equation in Higher Dimensions

讨论n维波动方程的Cauchy问题utt-Δu=0t=0∶u=φ(x),ut=ψ(x) t∈R,x=(x1,x2,…,xn)∈Rn的解,何时为T 周期的.设上述问题的解为u=u(x,t;φ,ψ),利用对部分变量作球平均的方法,籍助于归纳法,证明u(x,t;φ,ψ)为T 周期的充要条件是u(x,t;φ,0)与u(x,t;0,ψ)均为T 周期的.并据此给出了在n=5,4时,为使u(x,t;φ,ψ)为T 周期的,初始数据φ与ψ应满足的充分必要条件.

The author deals with the existence of time periodic solutions to the following Cauchy problem of wave equation in n dimensions:utt-Δu=0t=0∶u=φ(x),ut=ψ(x)t∈R, x=(x1,x2,...,xn)∈Rn.Under the assumption of the solution u=u(x,t;φ,ψ), by use of the method of spherical means to part of variables and induction, it can be given that the necessary and sufficient condition on u(x,t;φ,ψ)'s being time periodic is that u=u(x,t;φ,0) and u=u(x,t;0,ψ) are both time periodic. Given n=5 or 4, the author gives some necessary and sufficient conditions on the initial conditions φ and ψ, which guarantee that u(x,t;φ,ψ) is time periodic.