Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation

TL;DR

This paper investigates the behavior of solutions near the blowup threshold in the one-dimensional focusing nonlinear Klein-Gordon equation, revealing how solutions are trapped by a static solution depending on the power parameter.

Contribution

It combines numerical and analytical methods to analyze the trapping dynamics near the blowup threshold, highlighting the dependence on the nonlinearity power lphand extending previous rigorous results.

Findings

Solutions are trapped by the static solution S near the threshold.

Convergence to S is fast for lphand slow for lpha=1.

Convergence is very slow or absent for 0<lpha<1.

Abstract

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2\alpha} u =0$ on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution $S$ for intermediate times. The details of trapping are shown to depend on the power $\alpha$, namely, we observe fast convergence to $S$ for $\alpha>1$, slow convergence for $\alpha=1$, and very slow (if any) convergence for $0<\alpha<1$. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for $\alpha>2$) by Krieger, Nakanishi, and Schlag \cite{kns}.