Damped DNLW

The damped DNLW

$\Box u=u^{2}u_{t}$

in three dimensions is known to be locally well-posed in any sub-critical regularity $s>1$, and has scattering in $H^{3}$ Smh-p. It would be interesting to see whether one has local well-posedness in the critical energy regularity $H^{1}$. It is an example of a linear-derivative nonlinear wave equation.