Quadratic NLW/NLKG

[Thanks to Chengbo Wang for some corrections \u2013 Ed.]

• Scaling is $s_{c}=d/2-2$.
• For $d>4$ LWP is known for $s\geq d/2-2$ by Strichartz estimates LbSo1995. This is sharp by scaling arguments.
• For $d=4$ LWP is known for $s\geq 1/4$ 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 ([#two-speed see Overview]) one can improve this to $s>-1/4$ for non-linearities of the form $F=uv$ and $G=uv$ [Tg-p].

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