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 $s\geq 5/12$ 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 $L_{x}^{4}$). Below this regularity one cannot even make sense of the solution as a distribution.