Quartic NLW/NLKG

Scaling is $s c = d / 2 - 2 / 3$ .
For $d > 2$ LWP is known for $s \geq d/2 - 2/3$ by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases CtCoTa-p2
For d = 2 LWP is known for by Strichartz estimates. This is sharp by concentration arguments in the focusing case; the defocusing case is open.
- In the defocusing case one has GWP for $s > 2 / 3$ Fo-p
For $d = 1$ one has LWP for $s\geq 1/4$ by energy estimates and Sobolev (solution is in $L^4_x$ ). Below this regularity one cannot even make sense of the solution as a distribution.