The number of spatial critical points is nonincreasing in time, for positive, analytic solutions of a scalar, nonlinear, parabolic partial differential equation in one space dimension. While proving this, we answer the question: What happens to a critical point which loses simplicity?
(1989). Spatial Critical Points of Solutions of a One-Dimensional Nonlinear Parabolic Problem. Proceedings of the American Mathematical Society, 106 (4), 1003-1009.