具有Dirichlet边界条件的非线性波动方程的时间周期解  

Time Periodic Solutions of Nonlinear Waveequation with Dirichletboundary Conditions

作　　者：曾燕苗 马牧 

机构地区：[1]福州大学数学与统计学院,福建 福州

出　　处：《理论数学》2023年第4期1142-1150,共9页Pure Mathematics

摘　　要：本文考虑具有小振幅时间周期外力项的非线性波动方程的Dirichlet边值问题,关注其时间周期解的存在性。结合Lyapunov-Schmidt约化、隐函数定理和压缩映象原理,我们证明当外力频率满足一个特定的Diophantine型的非共振条件时,方程存在相同频率的时间周期解。进一步地,对所得的周期解,我们将建立其在Sobolev意义下与古典意义下的正则性结论。最后,我们还将给出在方程的静态平衡点附近周期解的局部唯一性。This paper is concerned with the time-periodic solutions of the Dirichlet boundary value problem for nonlinear wave equations in the presence of a time-periodic external forcing with frequency ω and amplitude ε. Combining the Lyapunov-Schmidt reduction, the implicit function theorem and the contraction mapping principle, we prove the existence of time-periodic solutions. The result holds for ω belongs to a Diophantine type parameter set. Moreover, we prove the regularity of the solutions in both Sobolev and classical cases. Finally, we prove the local uniqueness near a static equilibrium.

关 键 词：非线性波动方程 时间周期解 Lyapunov-Schmidt约化 

分 类 号：O17[理学—数学]