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A semilinear wave equation with slowly varying wave speed is considered inone 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 dividedby the wave speed and multiplied with the diameter of the spatial domain, is anadiabatic invariant: it remains nearly conserved over long times, longer thanany fixed power of the time scale of changes in the wave speed in the case ofone space dimension, and longer than can be attained by standard perturbationarguments in the two- and three-dimensional cases. The long-timenear-conservation of the action yields long-time existence of the solution. Theproofs use modulated Fourier expansions in time.

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