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 R3. 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 (Formula Presented). Furthermore, we show that the equilibrium can be stabilized with any rate less than (Formula Presented), provided (a, L) does not belong to a certain zero set. This rate is sharp.