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Boundary stabilization of the focusing NLKG equation near unstable equilibria: radial case

Joachim Krieger and Shengquan Xiang

Vol. 5 (2023), No. 4, 833–894
Abstract

We investigate the stability and stabilization of the cubic focusing Klein–Gordon equation around static solutions on the closed ball of radius L in 3. First we show that the system is linearly unstable near the static solution u 1 for any dissipative boundary condition ut + auν = 0,a (0,1). Then by means of open-loop boundary controls we stabilize the system around this equilibrium exponentially under the condition 2Ltan 2L. Furthermore, we show that the equilibrium can be stabilized with any rate less than 2 2L log 1+a 1a, provided (a,L) does not belong to a certain zero set. This rate is sharp.

Keywords
cubic Klein–Gordon, focusing, stabilization
Mathematical Subject Classification
Primary: 35B34
Milestones
Received: 12 September 2022
Revised: 30 January 2023
Accepted: 23 March 2023
Published: 15 December 2023
Authors
Joachim Krieger
Bâtiment des Mathématiques
EPFL
Lausanne
Switzerland
Shengquan Xiang
School of Mathematical Sciences
Peking University
Beijing
China