ORDER STRUCTURES AND THE HEAT-EQUATION

In this paper we shall introduce the notion of order structure and use it to study properties of a solution u(x, t) of a scalar linear parab olic equation. It is well known that in one space dimension the number of zeros of u(., t) is nonincreasing with t. This zero number is an e xample of an order structure and the zero number property is very usef ul in unfolding the structure of the global attractor of the semiflow generated by a scalar parabolic equation. We shall prove that in one s pace dimension the zero number is the only order structure preserved b y linear parabolic equations. In two dimensions the only order structu re preserved by linear parabolic equations is the order structure indu ced by the comparison principle for second order equations. Consequent ly, in two dimensions there does not exist a Fine decomposition into e quivalence classes as in the one dimensional case. (C) 1997 Academic P ress.