# Quartic NLW/NLKG

• Scaling is ${\displaystyle s_{c}=d/2-2/3}$.
• For ${\displaystyle d>2}$ LWP is known for ${\displaystyle s\geq d/2-2/3}$ by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases CtCoTa-p2
• For ${\displaystyle d=2}$ LWP is known for ${\displaystyle s\geq 5/12}$ 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 ${\displaystyle s>2/3}$ Fo-p
• For ${\displaystyle d=1}$ one has LWP for ${\displaystyle s\geq 1/4}$ by energy estimates and Sobolev (solution is in ${\displaystyle L_{x}^{4}}$). Below this regularity one cannot even make sense of the solution as a distribution.