On the decay estimate for small solutions to nonlinear Klein-Gordon equations with dissipative structure

TL;DR

This paper establishes decay estimates for small solutions to one-dimensional cubic nonlinear Klein-Gordon equations, demonstrating that solutions decay faster than free solutions when the nonlinearity has a dissipative structure.

Contribution

It provides new $L^p$-decay estimates for small solutions and shows enhanced decay rates due to dissipative nonlinearities.

Findings

Solutions decay faster than free solutions under dissipative structure

Established $L^p$-decay estimates for small data

Enhanced decay rates compared to non-dissipative cases

Abstract

We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the $L^p$-decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity has the suitable dissipative structure.