quasilinear PDE

A first-order PDE whose coefficients on the derivatives may depend on u itself, while the derivatives still appear only to the first power.

A first-order PDE is quasilinear when it has the form $a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)$ -- the coefficients $a,b$ may depend on $u$ itself, unlike a linear PDE, where they depend only on $x,y$ -- but $u_x$ and $u_y$ still enter linearly: never multiplied by each other, never raised to a power.