Boundary stabilization of the focusing NLKG equation 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 of radius L 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 under the condition $\sqrt{2}L\neq \tan \sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ \frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.