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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.

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Based upon a one-dimensional nonlinear Klein-Gordon equation with a perturbed one-gap periodic potential, this paper deals with the question as to whether spatially localized structures in periodic media can exist for all times. 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, . . .}. In this setting, the convergence to the trivial solution w. r. t. the L ∞ norm is shown in a canonical way. We obtain the corresponding linear rate. In contrast, if the spatially localized potential creates an eigenvalue in the band gap of the continuous spectrum, then we multiply u p by a spatial weight function and prove an asymptotic stability result w. r. t. a weighted L 2 norm for p ∈ {3, 4, 5, . . .}. Now, in the presence of an eigenvalue, there is a strongly reduced decay compared to the associated linearized problem. It is due to the component that belongs to the discrete spectral subspace of L 2 w. r. t. the perturbed Hill operator. As in [29], this phenomenon is referred to as metastability of the corresponding solutions.