an equation of mixed type

Imagine a road that is a smooth highway in one stretch and a rocky mountain track in another — you cannot drive it with a single technique; you must change gear where the terrain changes. A PDE of mixed type is just like this: it is one equation, but its type (elliptic, parabolic, or hyperbolic) is different in different parts of its domain, so no single solution method works everywhere. This is the natural consequence of allowing the coefficients to vary.

Recall that the type at a point is read off the discriminant B^2 - AC (in two variables) or the signs of the symbol's eigenvalues (in general). When the coefficients A, B, C depend on position, that quantity can change sign as you move around — and where it does, the type changes. The domain then splits into an elliptic region, a hyperbolic region, and a transition (parabolic, type-change) curve between them where the discriminant vanishes. The canonical example is the Tricomi equation y u_xx + u_yy = 0 (elliptic for y > 0, hyperbolic for y < 0, parabolic on y = 0); the Keldysh equation and the Chaplygin gas-dynamics equation are close relatives.

These equations are not just curiosities — they are exactly what governs transonic flow, where air over a wing is subsonic (elliptic) in some places and supersonic (hyperbolic) in others, with shock-laden behaviour at the sonic line. That is why they are unavoidable in aerodynamics. The honest difficulty is real: a mixed-type problem demands data appropriate to BOTH characters at once — boundary conditions where it is elliptic, Cauchy-like / characteristic conditions where it is hyperbolic — carefully matched across the transition line, and the well-posedness theory is far more delicate than for a pure type. Mixed-type problems are where the clean elliptic-parabolic-hyperbolic story meets the messy real world.