The parabolic equations of second order have only one family of real characteristics as case of b^2-4AC = 0 . For example, for the one-dimensional heat equation, the slope is
The characteristics are lines t = const . The perturbation that occurs at the space location x0 and time moment t0 (point P) affects the solution in the entire space domain 0 < x < L, although the effect becomes weaker with the distance to P. The domain of influence of the point P includes the domain 0 < x < L and times t >t(0). Accordingly, the domain of dependence of P includes all points 0 < x < L and all moments of time prior to the time of P.
In the physical systems described by parabolic equations, the perturbations are usually propagated by diffusion. The interaction occurs at infinite speed but relaxes with distance. For example, in the problem of conduction heat transfer in a one-dimensional rod, increase of temperature at a single point is initially felt only slightly at other points (weaker signal for larger distances). With time, however, the effect gradually becomes stronger and the nonuniformity of the temperature field is diffused or smoothed out.
Similar diffusive processes occur in other physical systems. For example, viscous terms in the Navier-Stokes equations (2.17) lead to
diffusion of gradients of the velocity field, thus giving the equations parabolic properties. The nonlinear parts of the material derivatives and the pressure effects can be neglected for some flows. The remaining equations such as
Steady-state viscous boundary layers is also types of parabolic equations