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1997 Spatial critical points of nonnegative solutions of the evolution $p$-Laplacian equation: the fast diffusion case
Shigeru Sakaguchi
Differential Integral Equations 10(6): 1049-1063 (1997).

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.

Citation

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Shigeru Sakaguchi. "Spatial critical points of nonnegative solutions of the evolution $p$-Laplacian equation: the fast diffusion case." Differential Integral Equations 10 (6) 1049 - 1063, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0940.35117
MathSciNet: MR1608025

Subjects:
Primary: 35J60
Secondary: 35B05

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 6 • 1997
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