论文标题
焦点NLKG在不稳定平衡附近的边界稳定：径向案例
Boundary stabilization of focusing NLKG near unstable equilibria: radial case
论文作者
论文摘要
我们研究了在$ \ mathbb {r}^3 $中，在半径l的闭合球上静态溶液周围的静态解决方案周围的静态解决方案的稳定性和稳定性。首先，我们证明该系统在任何耗散边界条件$ u_t+au_ν= 0，a \ in（0，1）$的静态解决方案附近是线性不稳定。然后，通过边界控制（开环和闭环），我们在条件$ \ sqrt {2} l \ neq \ tan \ tan \ sqrt {2} l $下稳定了该平衡的系统。此外，我们表明，可以用小于$ \ frac {\ sqrt {2}}} {2L} {2L} \ log {\ frac {1+a} {1-a}} $，提供$（a，l）$不属于某些ORE设置的任何速率稳定。这个速率很清晰。
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 $u_t+ au_ν=0, a\in (0, 1)$. 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.
