The Zero Number Diminishing Property under General Boundary Conditions

TL;DR

This paper extends the zero number diminishing property for solutions of one-dimensional parabolic equations to more general boundary conditions, including Robin and free boundary conditions, broadening its applicability in qualitative analysis.

Contribution

It generalizes the zero number diminishing property to problems with mixed boundary conditions, which was previously limited to Dirichlet conditions.

Findings

Zero number diminishes under general boundary conditions.

Applicable to Robin and free boundary problems.

Enhances qualitative analysis tools for parabolic equations.

Abstract

The so-called {\it zero number diminishing property} (or {\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary conditions, the number of zeroes of the solution $u(x,t)$ of a linear equation is finite, non-increasing and strictly decreasing when there are multiple zeroes (cf. \cite{Ang}). In this paper we extend the result to the problems with more general boundary conditions: $u= 0$ sometime and $u\not= 0$ at other times on the domain boundaries. Such results can be applied in particular to parabolic equations with Robin and free boundary conditions.