the parabolic type
Drop a spot of ink in still water, or press a hot coin onto a cold sheet of metal, then watch over time: the sharp spot softens, spreads, and blurs, smoothing relentlessly toward an even grey. Equations governing this kind of irreversible smoothing-as-time-passes are the parabolic type. They sit on the knife-edge exactly between the equilibrium ellipses and the propagating hyperbolas — and that in-between character shows in everything they do.
Formally, A u_xx + 2B u_xy + C u_yy + (lower order) = 0 is parabolic at a point when its discriminant B^2 - AC = 0 there, the borderline value. The defining archetype is the heat (diffusion) equation u_t = k u_xx. Read with x and t as the two variables, only u_xx among the second derivatives has a coefficient, so the principal part is degenerate — and that degeneracy is the type's fingerprint. There is exactly ONE real family of characteristic curves (here the lines t = constant), not the two of the hyperbolic case nor the none of the elliptic. Crucially, a parabolic equation is first order in the time variable, so it needs an initial condition in time PLUS boundary conditions in space.
The qualitative personality of a parabolic equation is unmistakable: instantaneous smoothing (rough or even discontinuous initial data becomes infinitely smooth the moment t > 0), infinite propagation speed (a localized heat source is felt, however faintly, everywhere at once — an honest idealization, since real heat does not outrun light), and irreversibility (you cannot run the film backwards; the backward heat equation is ill-posed, since reconstructing a sharp past from a blurred present amplifies the slightest noise without bound). The maximum principle and the steady-state limit (which is the associated elliptic equation) round out the picture.
The heat equation u_t = k u_xx on a rod, with the temperature given everywhere at t = 0 (initial condition) and held fixed at the two ends for all later times (boundary conditions), is the model parabolic problem: a unique solution that smooths and decays toward the steady state.
Parabolic = irreversible diffusion: initial data in time plus boundary data in space, one characteristic family, instant smoothing.
The single equal sign B^2 - AC = 0 is a borderline, not a comfortable region: any perturbation of the coefficients can tip the discriminant to either sign, and parabolic problems are genuinely well-posed only forward in time — running them backward is ill-posed.