Long-term analysis of semilinear wave equations with slowly varying wave speed
Ludwig Gauckler, Ernst Hairer and Christian Lubich
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.
Key Words. Semilinear wave equation, adiabatic invariant,
long-time existence, modulated Fourier expansion.