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 $\{y\} \times (0; T] $by a positive number M, then it has a powerlike decay of order $\frac{p}{2-p}$ with respect to the space variable x in$ R \times[ T/2; T].$ This fact , 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 eect is based on geometrical ideas.