Large data scattering for the defocusing NLKG on waveguide $\mathbb R^d\times\mathbb T$

TL;DR

This paper proves scattering for the defocusing nonlinear Klein-Gordon equation on a product space combining Euclidean and toroidal components, using profile decomposition and concentration-compactness techniques.

Contribution

It establishes scattering results for the defocusing NLKG on $ ^d imes $ in the energy subcritical case, extending understanding to mixed Euclidean-toroidal geometries.

Findings

Scattering holds for all initial data in the energy space for dimensions 1 to 4.

Develops a profile decomposition theorem in $ ^d imes $.

Employs concentration-compactness and rigidity methods for proof.

Abstract

We consider the pure-power defocusing nonlinear Klein-Gordon equation, in the energy subcritical case, posed on the product space $\mathbb R^d\times \mathbb T$, where $\mathbb T$ is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space $H^1 \times L^2$ for $1\leq d\leq 4$. The strategy consists in proving a suitable profile decomposition theorem in $\mathbb R^d\times \mathbb T$ to pursue a concentration-compactness \& rigidity method.