Quasilinear equations

Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not).

For instance, the following equations are all quasilinear (and the first two are NOT semilinear)

\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]

Mean curvature flow

\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]

Nonlocal porous medium equation

\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]

Hamilton-Jacobi with fractional diffusion

Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called Fully nonlinear equations, they include for instance the Monge Ampére Equation and Fully nonlinear integro-differential equations. Note that all Semilinear equations are automatically quasilinear.