The zero number diminishing property under general boundary conditions

Abstract

The so-called zero number diminishing property (or 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 zeros of the solution u(x,t) of a linear equation is finite, non-increasing and strictly decreasing when there are multiple zeros (cf. Angenent (1988)). In this paper we extend the result to the problems with more general boundary conditions: u=0 sometime and u≠0 at other times on the domain boundaries. Such results can be applied in particular to parabolic equations with Robin and free boundary conditions.