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 $s\geq 0$ 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 $F=uv$ and $G=uv$ (Tg-p).