标题： 非线性波动方程长期分析与缓慢变化的波速 显示英文标题

标题： Long-term analysis of semilinear wave equations with slowly varying wave speed

摘要： 考虑了一维到三维空间中在有界区间、矩形或盒子上的具有缓慢变化波速的半线性波动方程。 结果显示，作用量，即谐波能量除以波速并乘以空间区域的直径，是一个绝热不变量：在一位空间维度的情况下，它在比波速变化时间尺度的任何固定幂次更长的时间内几乎被保守，在二维和三维情况下，它在标准摄动论方法所能达到的时间范围之外几乎被保守。 作用量的长时间近似守恒导致了解的存在性。 证明使用了时间调制傅里叶展开。 显示英文摘要

摘要： A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly conserved over long times, longer than any fixed power of the time scale of changes in the wave speed in the case of one space dimension, and longer than can be attained by standard perturbation arguments in the two- and three-dimensional cases. The long-time near-conservation of the action yields long-time existence of the solution. The proofs use modulated Fourier expansions in time.