a parabolic operator

Think of how heat spreads through a metal bar, or how a drop of ink blurs into water. There is a single time direction in which things irreversibly even out, and at each instant the spreading is governed by a spatial smoothing law. A parabolic operator is the differential operator that packages exactly this kind of process: one privileged time derivative plus a 'diffusion-like' spatial part that pulls everything toward its average.

Concretely, the model is the operator P = d/dt + L acting on a function u(x, t), where L is a second-order elliptic operator in the space variables x — for the plain heat equation L = -Laplacian, so P u = u_t - Laplacian u. The equation P u = f, that is u_t + L u = f, is called a parabolic PDE. The defining feature is that the time derivative appears to first order while the space part L is elliptic (uniformly positive in the sense that its principal symbol never vanishes for nonzero spatial frequency). One special direction, time, is treated differently from all the space directions; that asymmetry is the whole personality of the equation. The general second-order linear case is u_t = sum a_ij(x,t) u_{x_i x_j} + (lower order), with the coefficient matrix a_ij positive definite.

Why single this class out? Because parabolic operators all share a family of strong, useful properties: solutions exist and are unique for an initial condition plus boundary conditions, they obey a maximum principle and comparison, and above all they smooth — rough initial data instantly becomes infinitely differentiable for t > 0. They also have infinite speed of propagation, so a localized disturbance is felt everywhere immediately, however faintly. This is the mature setting that the elementary heat equation is the first example of; the whole field studies what these operators do.