LONG-TIME ASYMPTOTICS OF THE ZERO LEVEL SET FOR THE HEAT EQUATION

Abstract

The zero level set Z(t) := {x is an element of R-d : u(x, t) = 0} of a solution u to the heat equation in R-d is considered. Under vanishing conditions on moments of the initial data, we will prove that the set Z(t) in a ball of radius C root t for any C > 0 converges to a specific graph as t -> infinity when the set is divided by root t. Solving a linear combination of the Hermite polynomials gives the graph, and coefficients of the linear combination depend on moments of the initial data. Also the graphs to which the zero level set Z(t) converges are classified in some cases.