Boundary Stabilization of Focusing NLKG near Unstable Equilibria: Radial Case

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 $u\equiv1$ for any dissipative boundary condition $u_t+au_ν=0,a∈(0,1)$. 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 $L≠tan\:L$.