## Differential and Integral Equations

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

Shigeru Sakaguchi

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 10, Number 6 (1997), 1049-1063.

Dates
First available in Project Euclid: 1 May 2013

https://projecteuclid.org/euclid.die/1367438218

Mathematical Reviews number (MathSciNet)
MR1608025

Zentralblatt MATH identifier
0940.35117

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Sakaguchi, Shigeru. Spatial critical points of nonnegative solutions of the evolution $p$-Laplacian equation: the fast diffusion case. Differential Integral Equations 10 (1997), no. 6, 1049--1063. https://projecteuclid.org/euclid.die/1367438218