Dynamics of threshold solutions for energy critical NLW with inverse square potential

We consider the focusing energy critical NLW with inverse square potential in dimensions d = 3, 4, 5. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifold of the ground state. In the latter case they converge to the ground state exponentially in the energy space as t -> infinity or t -> infinity. When the kinetic energy is greater than that of the ground state, we show that all solutions with finite mass blow up in finite time in both time directions in d = 3, 4. In d = 5, a finite mass solution can either have finite lifespan or lie on the stable/unstable manifolds of the ground state. The proof relies on the detailed spectral analysis of the linearized operator, local invariant manifold theory, and a global Virial analysis.

Automated summary

This article investigates the focusing energy critical nonlinear wave equation with inverse square potential in dimensions 3, 4, and 5. The characteristics of solutions on the energy surface of the ground state are analyzed. It is proven that solutions with kinetic energy lower than that of the ground state either scatter to zero or belong to the stable/unstable manifold of the ground state and converge exponentially to the ground state in the energy space as t -> infinity or t -> -infinity. When the kinetic energy is greater than that of the ground state, all solutions with finite mass blow up in finite time in dimensions 3 and 4. In dimension 5, a finite mass solution can either have a finite lifespan or lie on the stable/unstable manifolds of the ground state. The proof relies on detailed spectral analysis of the linearized operator, local invariant manifold theory, and global Virial analysis.