The Zero Number Diminishing Property under General Boundary Conditions

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.