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Dispersive estimates for solutions to the perturbed one‐dimensional Klein‐Gordon equation with and without a one‐gap periodic potential
Author(s) -
Prill Oskar
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200229
Subject(s) - mathematics , eigenfunction , norm (philosophy) , perturbation (astronomy) , mathematical analysis , dimension (graph theory) , operator (biology) , wave equation , space (punctuation) , stability (learning theory) , periodic potential , pure mathematics , eigenvalues and eigenvectors , quantum mechanics , physics , law , biochemistry , chemistry , linguistics , philosophy , repressor , machine learning , political science , transcription factor , computer science , gene
The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in [5][C. Blank, 2011] and [30][L. Tkeshelashvili, 2004]. Against this background, we consider a one‐dimensional linear Klein‐Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added. Given the dispersive character of the underlying equation, it is the purpose of this paper to deduce time‐decay rates for its solutions. We show that, generically, the part of the solution which is orthogonal to possible eigenfunctions of the perturbed Hill operator associated to the problem decays with a rate of t − 1 3w.r.t. the L ∞ norm. In weighted L 2 norms, we even get a time decay of t − 3 2. Furthermore, we consider the situation of a perturbing potential that is only made up of a spatially localized part which, now, can be slightly more general. It is well‐known that, in general, it is not possible to obtain the L ∞ endpoint estimate in one space dimension by means of the wave operators drawn from scattering theory. For this reason, we proceed directly and prove, along the lines of [17][M. Goldberg, 2004], the expected decay rate of t − 1 2.