a quasilinear first-order equation
A quasilinear first-order equation is one step wilder than a linear one. The first derivatives still appear linearly — no squares of u_x, no products of the two derivatives — but their coefficients are now allowed to depend on the unknown u itself. That single relaxation lets the equation feed back on its own solution, and it is the doorway to genuinely nonlinear behaviour like shocks.
The general form is a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u). Compare the linear case a(x,y) u_x + b(x,y) u_y + ... where the coefficients never see u; here a and b can. The method of characteristics still works, with the same system dx/ds = a, dy/ds = b, du/ds = c — but now the first two equations involve u, so the characteristic curves are no longer fixed in advance; their direction depends on the solution value they carry. You must integrate position and value together. Geometrically, the equation says the vector (a, b, c) is tangent to the solution surface at each point, so the surface is woven out of characteristic curves that bend according to the height u they reach.
Because the characteristics depend on u, fast ones can overtake slow ones and collide, and where they cross the smooth solution must break — this is exactly how a steepening wave forms a shock. Quasilinear equations are therefore the simplest setting where Burgers' equation, traffic-flow models, and conservation laws live, and the place where you first need weak solutions and an entropy condition to continue past the breakdown.
The traffic-flow equation rho_t + (Q(rho))_x = 0, written out, is rho_t + Q'(rho) rho_x = 0 — quasilinear, since the wave speed Q'(rho) depends on the car density rho. Denser traffic moves slower, so the back of a jam catches the front, characteristics cross, and a sharp jam front (a shock) forms.
Coefficients that depend on u make characteristics bend and collide — the birth of a shock.
Quasilinear is not the same as semilinear: in a semilinear equation only the lowest-order part (the right-hand side) sees u, while the top-derivative coefficients stay free of u; in quasilinear the coefficients of the derivatives themselves depend on u, which is what allows characteristics to cross.