Dispersive estimates for solutions to the perturbed one-dimensional Klein-Gordon equation with and without a one-gap periodic potential

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] and [30]. Against this background, we consider a one-dimensional linear Klein-Gordon equation to which both a spatially periodic Lame 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(-1/3) w.r.t. the L-infinity 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-infinity 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], the expected decay rate of t(-1/2). (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim