线性不稳定映射的非线性镇定
On nonlinear stabilization of linearly unstable maps
摘要：
我们研究了非线性稳定化的现象，展示了各种相关的例子和反例。对于Gâteaux可微映射，我们讨论了有限维和无限维的非线性稳定机制，特别适用于双曲型偏微分方程，并且，对于Fréchet可微映射的线性化算子是正常的，我们给出了一个尖锐的非线性指数不稳定性的线性速率的标准。这些结果突出了一个基本的开放性问题，即Fréchet可微性是否足以使线性指数不稳定性以可能较慢的速率暗示非线性指数不稳定性。
英文摘要：
We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.
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