Almost global solutions of 1D nonlinear Klein-Gordon equations with small weakly decaying initial data

TL;DR

This paper extends the understanding of 1D nonlinear Klein-Gordon equations by demonstrating the existence of almost global solutions with weakly decaying initial data, using dispersive estimates and phase analysis.

Contribution

It establishes almost global existence results for 1D Klein-Gordon equations with weak decay initial data, broadening the class of initial conditions for which solutions persist.

Findings

Solutions exist for longer times with small weighted Sobolev norm initial data.

Almost global solutions are obtained under weaker decay assumptions.

The proof employs dispersive estimates and phase function analysis.

Abstract

It has been known that if the initial data decay sufficiently fast at space infinity, then 1D Klein-Gordon equations with quadratic nonlinearity admit classical solutions up to time $e^{C/\epsilon^2}$ while $e^{C/\epsilon^2}$ is also the upper bound of the lifespan, where $C>0$ is some suitable constant and $\epsilon>0$ is the size of the initial data. In this paper, we will focus on the 1D nonlinear Klein-Gordon equations with weakly decaying initial data. It is shown that if the $H^s$-Sobolev norm with $(1+|x|)^{1/2+}$ weight of the initial data is small, then the almost global solutions exist; if the initial $H^s$-Sobolev norm with $(1+|x|)^{1/2}$ weight is small, then for any $M>0$, the solutions exist on $[0,\epsilon^{-M}]$. Our proof is based on the dispersive estimate with a suitable $Z$-norm and a delicate analysis on the phase function.