Semilinear

A semilinear equation is a PDE of the form

$Lu=F(u)$

where L is a linear operator and F is a nonlinear operator which does not involve any derivatives of u.

A semilinear-with-derivatives equation is a PDE of the form

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

where L is a linear operator, F is a nonlinear function of the first few derivatives $u,Du,\ldots ,D^{k}u$, with k strictly less than the order of L. Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than quasilinear or fully nonlinear equations.