Quasilinear

A quasilinear equation is an equation of the form

$F(u,Du,\ldots ,D^{k}u)=0$

which is linear (and nontrivial) in the top order terms $D^{k}u$. Thus a quasilinear equation takes the schematic form

$F(u,Du,\ldots ,D^{k-1}u)D^{k}u=G(u,Du,\ldots ,D^{k-1}u).$

By differentiating this equation up to $k-1$ times and working with the system of fields $v:=(u,Du,\ldots ,D^{k}u)$, one can place such equations in the slightly simpler form

$F(v)D^{k}v=G(v,Dv,\ldots ,D^{k-1}v)$

thoug hthis trick comes at the cost of lowering the regularity of the fields.

Quasilinear equations are more nonlinear than semilinear ones, but less nonlinear than fully nonlinear equations.