Global center stable manifold for the defocusing energy critical wave equation with potential

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 any solution scattering to $\phi$, which might be very far away from $\phi$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ 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 our previous result [18] to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from [3] 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 [14], 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 $\phi$.