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arXiv · 1012.1033

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

Piotr Bizoń·Tadeusz Chmaj·Nikodem Szpak
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Abstract

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2α} u =0$ 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 $α$, namely, we observe fast convergence to $S$ for $α>1$, slow convergence for $α=1$, and very slow (if any) convergence for $0<α<1$. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for $α>2$) by Krieger, Nakanishi, and Schlag \cite{kns}.

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Piotr Bizoń, Tadeusz Chmaj, Nikodem Szpak. 2011-01-16. Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation. https://doi.org/10.1063/1.3645363

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