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 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 $u_t+a{u}_{\nu }=0,a\in (0,1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $\sqrt{2}L\ne \tan \sqrt{2}L$. 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.

References in corpus (2)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Holomorphic families of forms, operators and $C_0$-semigroups