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

Gauckler, Ludwig; Hairer, Ernst; Lubich, Christian

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 longtime near-conservation of the action yields long-time existence of the solution. The proofs use modulated Fourier expansions in time.


Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2016
Sprache Englisch
Identifikator DOI(KIT): 10.5445/IR/1000052588
ISSN: 2365-662X
URN: urn:nbn:de:swb:90-525884
KITopen ID: 1000052588
Verlag KIT, Karlsruhe
Umfang 26 S.
Serie CRC 1173 ; 2016/6
Schlagworte semilinear wave equation, adiabatic invariant, long-time existence, modulated Fourier expansion
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