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 solutionu(x, t) of a scalar linear parabolic equation. It is well known that in one space dimension the number of zeros ofu(•, t) is nonincreasing witht. This zero number is an example of an order structure and the zero number property is very useful in unfolding the structure of the global attractor of the semiflow generated by a scalar parabolic equation. We shall prove that in one space dimension the zero number is the only order structure preserved by linear parabolic equations. In two dimensions the only order structure preserved by linear parabolic equations is the order structure induced by the comparison principle for second order equations. Consequently, in two dimensions there does not exist a fine decomposition into equivalence classes as in the one dimensional case.