Spatial critical points of nonnegative solutions of the evolution $p$-Laplacian equation: the fast diffusion case

Abstract

We consider the Cauchy problem for the fast diffusion equation: $$ \partial_t u = \text{{\rm div}}(|\nabla u|^{p-2}\nabla u) \text{ in } \mathbb{R}^N \times (0, \infty ) \text{ and } u(x,0) =\varphi (x) \text{ in } \mathbb{R}^N $$ with $ 1 <p < 2$ for nonzero bounded nonnegative initial data $\varphi$ having compact support, and show that the set of positive spatial critical points of the nonnegative solution is contained in the closed convex hull of the support of the initial datum for any time either when $\frac 32 < p <2$ and $N \geqq 2$ or when $1 < p <2$ and $N = 1$, and further in the case $ N = 1 $ it consists of one point after a finite time.