Spatial critical points of nonnegative solutions of the evolution \(p\)-Laplacian equation: the fast diffusion case
From author's abstract: We consider the Cauchy problem for the fast diffusion equation: \[ \partial _tu=\text{div} (|\nabla u|^{p-2}\nabla u) \text{ in } \mathbb{R}^n\times (0,\infty)\quad \text{and} \quad u(x,0)=\phi (x) \text{ in } \mathbb{R}^n \] with \( 1<p<2 \) for nonzero bounded nonnegative initial data \(\phi \) 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 \({3\over 2}<p<2\) and \(N\geq 2\) or when \(1<p<2\) and \(N=1,\) and further in the case \(N=1\) it consists of one point after a finite time.
compactly supported bounded nonnegative initial data
scientific article; zbMATH DE number 1450167