a quasilinear PDE

Push one notch deeper into the nonlinear world and you reach quasilinear equations — and this is where many of nature's most dramatic phenomena live, like waves that steepen and break. A quasilinear PDE is still linear in its highest derivatives, in the limited sense that those top derivatives appear only to the first power and are not multiplied by each other — but now their coefficients are allowed to depend on u itself and on its lower derivatives. The equation is 'linear if you freeze u', but u is part of the very rule that governs u, which feeds back in surprising ways.

The cleanest example is the inviscid Burgers equation u_t + u u_x = 0. The highest derivatives u_t and u_x each appear to the first power, so it is quasilinear — but the coefficient of u_x is u itself, the unknown. Read physically, the equation says each value of u travels to the right at a speed equal to its own height: tall parts of the wave move faster than short parts. So an initially smooth profile leans forward, steepens, and after a finite time the front becomes vertical — the derivative u_x blows up and a shock forms. No linear equation can do this; the feedback of u into its own transport speed is the whole story.

Quasilinear equations are the natural home of conservation laws (mass, momentum, energy in fluids and gases), traffic flow, and many geometric flows. Because the coefficient of the top derivative depends on the solution, even the equation's character — where it behaves like a wave versus where it might steepen — can change from place to place and from moment to moment as u evolves. This is why finite-time blow-up, shocks, and the need for weak solutions and entropy conditions all first appear here, and why first-order quasilinear equations are solved by the method of characteristics, following curves along which the equation collapses to an ODE.