# Cubic NLW/NLKG on R

• Scaling is ${\displaystyle s_{c}=-1/2}$.
• LWP for ${\displaystyle s\geq 1/6}$ by energy estimates and Sobolev (solution is in ${\displaystyle L_{x}^{3}}$).
• For ${\displaystyle s<1/6}$ one has ill-posedness (CtCoTa-p2), indeed it is not even possible to make sense of solutions in the distributional sense.
• GWP for ${\displaystyle s>1/3}$ for defocussing NLKG (Bo1999)
• For ${\displaystyle s\geq 1}$ 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 ${\displaystyle H^{1/2}}$. NLW is similar but with the homogeneous ${\displaystyle H^{1/2}}$.
• 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.