Quadratic NLW/NLKG

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  • Scaling is .
  • For LWP is known for by Strichartz estimates (LbSo1995). This is sharp by scaling arguments.
  • For LWP is known for by Strichartz estimates (LbSo1995).This is sharp from Lorentz invariance (concentration) considerations.
  • For LWP is known for by Strichartz estimates (LbSo1995).
    • One has ill-posedness for (Lb1996). This is related to the failure of endpoint Strichartz when .
  • For LWP is known for by Strichartz estimates (or energy estimates and Sobolev in the case).
    • For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity as a distribution (CtCoTa-p2).
    • In the two-speed case one can improve this to for non-linearities of the form and (Tg-p).