Abstract:

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi,0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It is constructed in a first step locally around {\it any solution scattering to} $\phi$, which might be very far away from $\phi$ in the $\dot H^1\times L^2(\Bbb{R}^3)$ norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends a previous result of ours to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from Beceanu-Goldberg to construct a local center stable manifold near any solution that scatters to $(\phi,0)$. This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle, which is used to show that solutions that start off but near the local manifolds associated with $\phi$ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state $\phi$.

摘要：

在本文中，我们考虑了散焦能量临界波方程，其俘获势的维数为$ 3 $。我们证明，解决方案散射到不稳定激发态$（\ phi，0）$的初始数据集形成了在能量空间中连接$ C ^ 1 $流形的有限维路径。该流形是与$（\ phi，0）$关联的全局且唯一的中心稳定流形。它是在{周围将任何解决方案分散到} $ \ phi $本地的第一步中构建的，它可能与$ \ dot H ^ 1 \ times L ^ 2（\ Bbb { R} ^ 3）$范数。在第二个关键步骤中，对于任何在附近但不是在本地歧管上开始的解决方案，都证明了不归属性。这样可以确保局部歧管形成整体。因此，散射到不稳定的稳态是一种非常规行为，在能量空间中具有强烈的拓扑意义。这将我们先前的结果扩展到非径向情况。这里的新成分是（i）应用Beceanu-Goldberg的反向Strichartz估计在分散到$（\ phi，0）$的任何解决方案附近构造局部中心稳定流形。这是必需的，因为标准Strichartz估计的端点非径向失败。（ii）Duyckaerts-Kenig-Merle引入的能量非径向估计通道，该通道用于表明，开始但接近$ \ phi $的局部流形的解向远场发出的能量超过超出稳态$ \ phi $的能量。这里的新成分是（i）应用Beceanu-Goldberg的反向Strichartz估计在分散到$（\ phi，0）$的任何解决方案附近构造局部中心稳定流形。这是必需的，因为标准Strichartz估计的端点非径向失败。（ii）Duyckaerts-Kenig-Merle引入的非径向能量估算通道，该通道用于显示开始但接近$ \ phi $的局部流形的溶液向远场发出的能量超过超出稳态$ \ phi $的能量。这里的新成分是（i）应用Beceanu-Goldberg的反向Strichartz估计在分散到$（\ phi，0）$的任何解决方案附近构造局部中心稳定流形。这是必需的，因为标准Strichartz估计的端点非径向失败。（ii）Duyckaerts-Kenig-Merle引入的能量非径向估计通道，该通道用于表明，开始但接近$ \ phi $的局部流形的解向远场发出的能量超过超出稳态$ \ phi $的能量。