Fast/slow diffusion and collapsing sandpiles,
type: Item
instance of: scholarly article
title: Fast/slow diffusion and collapsing sandpiles,
zbMATH Open document ID: 0879.35019
published in: Journal of Differential Equations
publication date: 1998-01-29T00:00:00Z
review text: The authors study the behaviour of solutions \(u_p\) of the initial value problem \[ u_{p,t}- \Delta_pu_p= 0\quad\text{in }\mathbb{R}^n\times (0,\infty),\quad u_p=g\quad\text{in }\mathbb{R}^n\times\{t= 0\}, \] \(1\leq p<\infty\), in the infinitely fast/infinitely slow diffusion limit \(p\to\infty\). Here \(\Delta_p\) is the \(p\)-Laplacian and \(g:\mathbb{R}^n\to\mathbb{R}\) denotes a compactly supported Lipschitz function with Lipschitz constant \(>1\). It is shown that \(u_p(\cdot,t)\) converges uniformly as \(p\to\infty\) to a time-independent function \(u\) such that \(|Du|_{L^\infty}\leq 1\). This can be interpreted as a model for the collapse of a sandpile from an initially unstable configuration. The function \(g\) determines the height of the sandpile at \(t=0\), \(u\) describes the stable profile after the motion ceases. It should be noted that in certain cases explicit solutions can be calculated.
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1040628
zbMATH Keywords: unstable sandpiles
zbMATH Keywords: \(p\)-Laplacian
author: Lawrence C. Evans
author: Mikhail Feldman
author: Ronald F. Gariepy
reviewed by: M. Fuchs
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3243
OpenAlex ID: W1994571215
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DOI: 10.1006/JDEQ.1997.3243
description: scientific article; zbMATH DE number 1040628