To define the simplest kind of parabolic PDE, consider a real-valued function of two independent real variables, and. A second-order, linear, constant-coefficient PDE for takes the form and this PDE is classified as being parabolic if the coefficients satisfy the condition Usually represents one-dimensional position and represents time, and the PDE is solved subject to prescribed initial and boundary conditions. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation to define a planar parabola. The basic example of a parabolic PDE is the one-dimensional heat equation, where is the temperature at time and at position along a thin rod, and is a positive constant. The symbol signifies the partial derivative of with respect to the time variable, and similarly is the second partial derivative with respect to. For this example, plays the role of in the general second-order linear PDE: ,, and the other coefficients are zero. 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 measures how far off the temperature is from satisfying the mean valueproperty ofharmonic 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. This equation is the prototype of a multi-dimensional parabolic PDE. Noting that is an elliptic operator suggests a broader definition of a parabolic PDE: where is a second-order elliptic operator . A system of partial differential equations for a vector can also be parabolic. For example, such a system is hidden in an equation of the form if the matrix-valued function has a kernel of dimension 1. Parabolic PDEs can also be nonlinear. For example, Fisher's equation is a nonlinear PDE that includes the same diffusion term as the heat equation but incorporates a linear growth term and a nonlinear decay term.
Solution
Under broad assumptions, an initial/boundary-value problem for a linear parabolic PDE has a solution for all time. The solution, as a function of for a fixed time, is generally smoother than the initial data. For a nonlinear parabolic PDE, a solution of an initial/boundary-value problem might explode in a singularity within a finite amount of time. It can be difficult to determine whether a solution exists for all time, or to understand the singularities that do arise. Such interesting questions arise in the solution of the Poincaré conjecture via Ricci flow.
One occasionally encounters a so-called backward parabolic PDE, which takes the form . An initial-value problem for the backward heat equation, is equivalent to a final-value problem for the ordinary heat equation, An initial/boundary-value problem for a backward parabolic PDE is usually not well-posed . Nonetheless, these problems are important for the study of the reflection of singularities of solutions to various other PDEs. Moreover, they arise in the pricing problem for certain financial instruments.
