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

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx}+u-\mathrm{\mid }u{\mathrm{\mid }}^{2\alpha }u=0$ 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}.