Boundary stabilization of focusing NLKG near unstable equilibria: radial case - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2020

Boundary stabilization of focusing NLKG near unstable equilibria: radial case

Abstract

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in $\mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $u\equiv 1$ for any dissipative boundary condition $u_t+ au_{\nu}=0, a\in (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially with rate less than $ \frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$, which is sharp, provided that the radius of the ball $L$ satisfies $L\neq \tan L$.

Dates and versions

hal-03053740 , version 1 (11-12-2020)

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Joachim Krieger, Shengquan Xiang. Boundary stabilization of focusing NLKG near unstable equilibria: radial case. 2020. ⟨hal-03053740⟩
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