Asymptotic profiles of zero points of solutions to the heat equation

TL;DR

This paper investigates the long-term behavior of zero points in solutions to the heat equation, revealing new asymptotic profiles and extending previous results to higher dimensions and orders.

Contribution

It introduces a novel asymptotic expansion theory for zero points of the heat equation, including divergence at O(t) and higher-order profiles in multiple dimensions.

Findings

Zero points diverge at a rate of O(t).

Second and third-order asymptotic profiles are characterized in 1D.

Results extend to high-dimensional spaces for zero-level sets.

Abstract

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at $O(t)$. In a one-dimensional spatial case, we show the zero point's second and third-order asymptotic profiles in a general situation. We also analyze a zero-level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case.