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 ${ℝ}^3$. First we show that the system is linearly unstable near the static solution $u\equiv 1$ for any dissipative boundary condition $u_t+a{u}_{\nu }=0,a\in (0,1)$. Then by means of open-loop boundary controls we stabilize the system around this equilibrium exponentially under the condition $\sqrt{2}L\ne \tan <!--FUNCTION\; APPLICATION-->\sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $\frac{\sqrt{2}}{2L}\log <!--FUNCTION\; APPLICATION-->\frac{1+a}{1-a}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.

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