Cubic NLW/NLKG on R4

• Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 1} .
• LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1} by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
  • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1} the time of existence depends on the profile of the data and not just on the norm.
  • One has strong uniqueness in the energy class Pl-p5, FurPlTer2001. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s<s_c} one has instantaneous blowup in the focusing case, and unbounded growth of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} norms in the defocusing case (CtCoTa-p2).
• GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1} in the defocussing case SaSw1994 (see also Gl1990, Gl1992, Sw1988, Sw1992, BaSa1998, BaGd1997).
  • In the focussing case there is blowup from large data by the ODE method.