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Asymptotic stability near the soliton for quartic Klein–Gordon equation in 1D

Adilbek Kairzhan and Fabio Pusateri

Vol. 5 (2023), No. 4, 795–832
Abstract

We consider the nonlinear focusing Klein–Gordon equation in 1 + 1 dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay, and local decay, for even perturbations of the static soliton originating from well-prepared initial data belonging to a subset of the center-stable manifold constructed by Bates and Jones (1989) and Kowalczyk, Martel and Muñoz (2022). Our results complement those of Kowalczyk, Martel and Muñoz and confirm numerical results of Bizoń, Chmaj and Szpak (2011) when considering nonlinearities up with p 4. In particular, we provide new information both local and global in space about asymptotically stable perturbations of the soliton under localization assumptions on the data.

Keywords
asymptotic stability, Klein–Gordon equation, weighted spaces
Mathematical Subject Classification
Primary: 35P25, 35Q55, 42B37, 43A32
Milestones
Received: 18 July 2022
Revised: 22 June 2023
Accepted: 23 August 2023
Published: 15 December 2023
Authors
Adilbek Kairzhan
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Fabio Pusateri
Department of Mathematics
University of Toronto
Toronto, ON
Canada