The Threshold Energy For Focusing Nonlinear Klein-Gordon Equations On Hyperbolic Spaces

This paper considers focusing subcritical nonlinear Klein-Gordon equations on hy-perbolic spaces, and applys variational estimates to prove that when the energy of initial data lie below the threshold energy, the energy space corresponding to initial data and solution could be divided into two parts. Initial data belonging to one part of the energy space lead to finite time blow-up of solution, while initial data belonging to the other part lead to global well-posedness of solution in time. That is to say, the dichotomy between finite time blow-up and global well-posedness is obtained. In order to prove the main result, this paper defines a variety of energy functionals, and proves variational estimates corresponding to these functionals. These variational estimates play crucial roles in the proof of the dichotomy.