Cubic NLW/NLKG on R

• Scaling is $s_{c}=-1/2$.
• LWP for $s\geq 1/6$ by energy estimates and Sobolev (solution is in $L_{x}^{3}$).
  • 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.