@article{22022,
  abstract     = {We consider both the defocusing and focusing cubic nonlinear Klein–Gordon equations
utt − Δu + u ± u3 =0
in two space dimensions for real-valued initial data u(0) ∈ H1/x and ut(0) ∈ L2/x.
We show that in the defocusing case, solutions are global and have finite global L4/t,x spacetime bounds. In the focusing case, we characterize the dichotomy
between this behaviour and blowup for initial data with energy less than that
of the ground state.
These results rely on analogous statements for the two-dimensional cubic
nonlinear Schr¨odinger equation, which are known in the defocusing case and
for spherically-symmetric initial data in the focusing case. Thus, our results
are mostly unconditional.
It was previously shown by Nakanishi that spacetime bounds for Klein–
Gordon equations imply the same for nonlinear Schr¨odinger equations.},
  author       = {Killip, Rowan and Stovall, Betsy and Visan, Monica},
  issn         = {1088-6850},
  journal      = {Transactions of the American Mathematical Society},
  number       = {3},
  pages        = {1571--1631},
  publisher    = {American Mathematical Society},
  title        = {{Scattering for the cubic Klein-Gordon equation in two space dimensions}},
  doi          = {10.1090/s0002-9947-2011-05536-4},
  volume       = {364},
  year         = {2012},
}