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Definition

To define the simplest kind of parabolic PDE, consider a real-valued function u (x, y) {\displaystyle u(x,y)} of two independent real variables, x {\displaystyle x} and y {\displaystyle y}. A second-order, linear, constant-coefficient PDE for u {\displaystyle u} takes the form

where the subscripts denote the first- and second-order partial derivatives with respect to x {\displaystyle x} and y {\displaystyle y}. The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of u {\displaystyle u}) satisfy the condition

Usually x {\displaystyle x} represents one-dimensional position and y {\displaystyle y} represents time, and the PDE is solved subject to prescribed initial and boundary conditions. Equations with B 2 − A C < 0 {\displaystyle B^{2}-AC<0} are termed elliptic while those with B 2 − A C > 0 {\displaystyle B^{2}-AC>0} are hyperbolic. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation A x 2 + 2 B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+2Bxy+Cy^{2}+Dx+Ey+F=0} to define a planar parabola.

The basic example of a parabolic PDE is the one-dimensional heat equation

where u (x, t) {\displaystyle u(x,t)} is the temperature at position x {\displaystyle x} along a thin rod at time t {\displaystyle t} and α {\displaystyle \alpha } is a positive constant called the thermal diffusivity.

The heat equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity u x x {\displaystyle u_{xx}} measures how far off the temperature is from satisfying the mean value property of harmonic functions.

The concept of a parabolic PDE can be generalized in several ways.For instance, the flow of heat through a material body is governed by the three-dimensional heat equation

where

denotes the Laplace operator acting on u {\displaystyle u}. This equation is the prototype of a multi-dimensional parabolic PDE.

Noting that − Δ {\displaystyle -\Delta } is an elliptic operator suggests a broader definition of a parabolic PDE:

where L {\displaystyle L} is a second-order elliptic operator (implying that L {\displaystyle L} must be positive;a case where u t = + L u {\displaystyle u_{t}=+Lu} is considered below).

A system of partial differential equations for a vector u {\displaystyle u} can also be parabolic.For example, such a system is hidden in an equation of the form

if the matrix-valued function a (x) {\displaystyle a(x)} has a kernel of dimension 1.