Quadratic NLW/NLKG

Scaling is $s_c = \frac{d}{2} - 2$ .
For $d > 4$ LWP is known for $s \geq \frac{d}{2} - 2$ by Strichartz estimates (LbSo1995). This is sharp by scaling arguments.
For $d = 4$ LWP is known for $s \geq \frac{1}{4}$ by Strichartz estimates (LbSo1995).This is sharp from Lorentz invariance (concentration) considerations.
For d = 3 LWP is known for s > 0 by Strichartz estimates (LbSo1995).
- One has ill-posedness for $s = 0$ (Lb1996). This is related to the failure of endpoint Strichartz when $d = 3$ .
For d = 1,2 LWP is known for by Strichartz estimates (or energy estimates and Sobolev in the d = 1 case).
- For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity $f 2$ as a distribution (CtCoTa-p2).
- In the two-speed case one can improve this to $s > - 1 / 4$ for non-linearities of the form $F = u v$ and $G = u v$ (Tg-p).