Asymptotics and scattering for wave Klein-Gordon systems

Abstract

AbstractWe study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a slightly simplified version of global existence based on LeFloch-Ma, and then derive the asymptotic behavior of the system. The asymptotics of the Klein-Gordon field consist of a modified phase times a homogeneous function, and the asymptotics of the wave equation consist of a radiation field in the wave zone and an interior homogeneous solution coupled to the Klein-Gordon asymptotics. We then consider the inverse problem, the scattering from infinity. We show that given the type of asymptotic behavior at infinity, there exist solutions of the system that present the exact same behavior.KEYWORDS: AsymptoticsScattering from infinityWave-Klein-Gordon systems Notes1 In fact, u3 would be zero if we only consider the term ϕ02. The presence of u3 comes from the lower order terms in (∂tϕ0)2.2 We use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean.3 Here we use the decay |△yϕ|≲ερ−32+δ(1−|y|2)74−δ, for which we only have it with 47 replaced by 45 at this stage, but this can also be shown by dealing with commutators like the existence proof.4 Recall in Lemma 4.5, we require that α cannot be too worse. i.e. close to –1. However, once α satisfies this condition, the value of α does not affect the outcome of the estimate.5 Note that here we take the contribution from (∂tϕ)2 into account, which we did not consider in the introduction part for simplicity.Additional informationFundingH.L. was supported in part by Simons Collaboration Grant 638955. X.C. thanks Junfu Yao for helpful discussions.