On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials

TL;DR

This paper studies the long-term behavior of small solutions to a 1D Klein-Gordon equation with variable coefficients and a non-generic potential, revealing a novel modified scattering involving logarithmic decay slowdown caused by threshold resonance.

Contribution

It extends previous work by analyzing modified scattering in the presence of a non-generic potential and variable coefficients, highlighting the role of threshold resonance.

Findings

Discovered a modified scattering phenomenon with logarithmic decay slowdown.

Linked the phenomenon to the threshold resonance of the linear operator.

Extended understanding of asymptotic stability in scalar field theories.

Abstract

We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein-Gordon operator. It was previously uncovered for the special case of the zero potential in [51]. The Klein-Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line.