聚焦NLKG在非稳定平衡点附近的边界镇定：径向情形
Boundary stabilization of focusing NLKG near unstable equilibria: radial case
AI总结：
研究聚焦三次Klein-Gordon方程在闭球上静态解附近的线性不稳定性和边界镇定，提出开环与闭环控制实现指数镇定，并给出最优镇定速率。
摘要：
我们研究了在$\u211d^3$中半径为$L$的闭球上，三次聚焦Klein-Gordon方程在静态解附近的稳定性与镇定问题。首先，我们证明对于任何耗散边界条件$u_t+ a u_\nu=0, a\in (0, 1)$，系统在静态解$u\equiv 1$附近是线性不稳定的。然后，通过边界控制（开环和闭环），我们在条件$\sqrt{2}L\neq \tan \sqrt{2}L$下，以指数速率将系统镇定在该平衡点附近。此外，我们证明，只要$(a,L)$不属于某个零集，该平衡点可以以任何小于$\frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$的速率被镇定。这个速率是精确的。
英文摘要：
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.
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