1-dimensional Harnack estimates

Let u be a non-negative super-solution to a 1-dimensional singular parabolic equation of p-Laplacian type (1 < p < 2). If u is bounded below on a time-segment fyg-y} (0, T] by a positive number M, then it has a powerlike decay of order p/2-p with respect to the space variable x in ℝ × [T/2, T]. This fact, stated quantitatively in Proposition 1.2, is a "sidewise spreading of positivity" of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.