Zero Set of the Solution to One-Dimensional Parabolic Equation

Abstract

In this paper, we investigate the one-dimensional parabolic equation. Using blow-up analysis, we prove that the zero set at each time slice is discrete and that the number of zeros decreases with respect to time. Moreover, it strictly declines at the singularity. This provides a new and simplified proof of the main results in existing literature, where the original proof relies on spectral theory. Besides, our results represent a localized version of those existing results with weaker conditions.