Linear Parabolic Operators

Abstract

The flow of a quantity, such as heat, is described by the diffusion equation, where the total amount of the quantity does not change. Thus, both heat and mass are conserved, although their distributions may undergo slight changes. The first order time derivative in the diffusion equation signifies a complete determination of the time development of the solution subject to prescribed initial temperature or mass distribution, and presence of any sources or sinks. It also means the irreversibility of the equation in the sense that the solution is always in the direction of time; thus, a change in the sign of time also changes the behavior of the solution. The fundamental solutions for the parabolic operator play an important role in problems of heat conduction (§13.2) and neutron diffusion (§13.4).