Second-Order Linear Parabolic Equations

Abstract

Parabolic equations arise in diffusion processes, and more generally in "irreversible" time-dependent processes. Mathematically, this is reflected in the fact that the equations are not invariant under the reversal of time ; i.e., under the transformation t-t. This means that knowledge about the "past" is lost as time increases. For example, there may be dissipation effects which lead to an increase in entropy and a consequent loss of information. The simplest example of a parabolic equation is the equation of "heat conduction" Hu == Ut-ki1u = 0, (9.1) where i1 is the Laplace operator, and k is a positive constant. It is quite clear that this equation changes form when t is replaced by-t. Observe however, that (9.1) is preserved under the transformation (x, t) _ «(Xx, (X2t). This transformation also leaves invariant the expression 1 x 12 It; indeed we shall see that this latter expression plays an important role in the study of (9.1). We shall show that solutions of parabolic equations obey a maximum principle, and that parabolic operators are "smoothing," in the sense that the solutions are more smooth than the data. §A. The Heat Equation We consider the equation (9.1) in a cylindrical region of x-t space of the form qJJ = R x (0, T), T< 00, where R is a bounded region in Rn. We let qJJ' = cl(R) x {t = O} u oR x [0, TJ;