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

Abstract

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 and . 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−13 w.r.t. the L∞ norm. In weighted L2 norms, we even get a time decay of t−32. 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 , the expected decay rate of t−12.

Asymptotic stability of the vacuum solution for one‐dimensional nonlinear Klein‐Gordon equations with a perturbed one‐gap periodic potential with and without an eigenvalue

“…In fact, B 2 is a strictly positive self-adjoint operator defined on the Hilbert space L 2 with domain W 2,2 . More details can be found in [23]. Pertaining to the uniform estimate, we only need the Cauchy-Schwarz inequality.…”

“…On the one hand, we need dispersive estimates of the corresponding linearized problem. They can be drawn from [23] where the large-time behavior of solutions to Cauchy problem (I) with λ = 0 is studied. On the other hand, singular resolvent estimates are useful to treat those terms that are still left after the key resonance has been isolated by an integration by parts.…”

“…That is why the first part of this paper is devoted to the proof of the necessary singular resolvent estimates. The linear decay estimates can be drawn from [23].…”

“…Mathematically, the goal is to determine the time-asymptotic behavior of small-amplitude solutions to (I) for the situations where the discrete spectrum of the operator B 2 := −∂ 2 x + V (x) + μ contains either no or exactly one eigenvalue. For this reason, we apply our results in [23] where decay estimates for the corresponding linear problem are proved. Depending on the spectrum of the operator B 2 , the role of the nonlinearity changes: Whereas, in the case of a purely absolutely continuous spectrum, it may be considered as a perturbation of the associated linearized equation, the nonlinear term becomes indispensable in the presence of an eigenvalue by providing the mechanism of radiation damping.…”

“…As it turns out that, given our model equation, the latter question cannot be answered in the affirmative, we show the asymptotic stability of the vacuum state in appropriate dispersive norms and provide an upper bound for the temporal decay rates of the corresponding solutions. This is done by using the dispersive estimates proved in [23]. More precisely, if the perturbed Hill operator associated to our problem has no eigenvalue, we add a power nonlinearity u p with p ∈ {6, 7, 8, .…”