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
DOI: 10.12677/PM.2023.134119, PDF,    国家自然科学基金支持
作者: 曾燕苗, 马 牧*：福州大学数学与统计学院，福建 福州
关键词: 非线性波动方程；时间周期解；Lyapunov-Schmidt约化；Nonlinear Wave Equation； Time-Periodic Solution； Lyapunov-Schmidt Reduction
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。
Abstract: 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.
文章引用：曾燕苗, 马牧. 具有Dirichlet边界条件的非线性波动方程的时间周期解[J]. 理论数学, 2023, 13(4): 1142-1150. https://doi.org/10.12677/PM.2023.134119

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