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

TL;DR

This paper demonstrates that the action in semilinear wave equations with slowly varying wave speed remains nearly conserved over long times, ensuring long-term solution existence across one to three dimensions using modulated Fourier expansions.

Contribution

It establishes the adiabatic invariance of the action for semilinear wave equations with slowly varying wave speed in multiple dimensions, extending long-time conservation results.

Findings

Action is nearly conserved over long times in 1D, 2D, and 3D cases.

Long-time existence of solutions is proved based on this invariance.

Modulated Fourier expansions are used to establish the results.

Abstract

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.