Order Structures and the Heat Equation

In this paper we shall study the evolution of the zero set of the solution of the heat equation perturbed by a potential c and Neumann boundary conditions in one and two dimensions. 1. Introduction Let IR d be a bounded connected domain with a smooth boundary @ (we assume throughout that is of class C 3) and let c be a continuous function on , the closure of. Consider the heat equation with a potential c and Neumann boundary conditions (u t (x; t) = u(x; t) + c(x)u(x; t); for x 2 ; t > 0; u n (x; t) = 0; for x 2 @; t > 0;