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Abstract
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We consider the nonlinear focusing Klein–Gordon equation in
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
with
. 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.
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Keywords
asymptotic stability, Klein–Gordon equation, weighted
spaces
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Mathematical Subject Classification
Primary: 35P25, 35Q55, 42B37, 43A32
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Milestones
Received: 18 July 2022
Revised: 22 June 2023
Accepted: 23 August 2023
Published: 15 December 2023
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Publishers). |
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