Generic and non-generic behavior of solutions to the defocusing energy critical wave equation with potential in the radial case

Abstract

In this paper, we continue our study [16] on the long time dynamics of radial solutions to defocusing energy critical wave equation with a trapping radial potential in 3 + 1 dimensions. For generic radial potentials (in the topological sense) there are only finitely many steady states which might be either stable or unstable. We first observe that there can be stable excited states (i.e., a steady state which is not the ground state) if the potential is large and attractive, although all small excited states are unstable. We prove that the set of initial data for which solutions scatter to any one unstable excited state forms a finite co-dimensional connected $C^1$ manifold in energy space. This amounts to the construction of the global path-connected, and unique, center-stable manifold associated with, but not necessarily close to, any unstable steady state. In particular, the set of data for which solutions scatter to unstable states has empty interior in the energy space, and generic radial solutions scatter to one of the stable steady states. Our main tools are (i) near any given finite energy radial initial data $(u_0,u_1)$ for which the solution $u(t)$ scatters to some unstable steady state $\phi$ we construct a $C^1$ manifold containing $(u_0,u_1)$ with the property that any solution starting on the manifold scatters to $\phi$; moreover, any solution remaining near the manifold for all positive times lies on the manifold (ii) an exterior energy inequality from [9,10,16]. The latter is used to obtain a result in the spirit of the one-pass theorem [22], albeit with completely different techniques