Cubic NLW/NLKG on R

Scaling is $s c = - 1 / 2$ .
LWP for by energy estimates and Sobolev (solution is in ).
- For $s < 1 / 6$ one has ill-posedness (CtCoTa-p2), indeed it is not even possible to make sense of solutions in the distributional sense.
GWP for s > 1 / 3 for defocussing NLKG (Bo1999)
- For $s \geq 1$ this is clear from energy conservation (for both NLKG and NLW).
- Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
- In the focussing case there is blowup from large data by the ODE method.
Remark: NLKG can be viewed as a symplectic flow with the symplectic form of $H 1 / 2$ . NLW is similar but with the homogeneous $H 1 / 2$ .
Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see De2001, LbSf-p.