Septic NLW/NLKG on R

• Scaling is $s_{c}=1/6$.
• LWP for $s\geq 5/14$ by energy estimates and Sobolev (solution is in $L_{x}^{7}$)
  • For $s<5/14$ it is not even possible to make sense of the solution as a distribution (note there are no smoothing effects for the 1D wave equation!) and so there is ill-posedness [CtCoTa-p2].
• GWP for $s=1$ in the defocusing case by energy conservation. Presumably this is improvable.