Linear, semilinear, quasilinear, and fully nonlinear partial differential equations

Consider an order-$m$ scalar PDE on $\mathrm{\Omega }$.

It is linear when it has the form

${\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{a}_{\alpha }(x)D^{\alpha }u(x)=f(x),$

so every jet variable enters affinely and with coefficients depending only on $x$.

It is semilinear when the top-order derivatives enter linearly with coefficients depending only on $x$, while lower-order terms may depend nonlinearly on $(u,Du,\dots ,D^{m-1}u)$.

It is quasilinear when the top-order derivatives still enter linearly, but their coefficients may depend on $(x,u,Du,\dots ,D^{m-1}u)$.

It is fully nonlinear when the dependence on the order-$m$ jet is not affine. Thus semilinear and quasilinear equations are affine in the top-order variables after the lower jet is fixed, whereas fully nonlinear equations are not. The classification is by dependence on the highest-order jet, not by how the equation happens to be typeset.