Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian
type: Item
instance of: scholarly article
title: Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian
zbMATH Open document ID: 1190.35127
author: Goro Akagi
author: Petri Juutinen
author: Ryuji Kajikiya
published in: Mathematische Annalen
publication date: 2009-03-26T00:00:00Z
review text: This paper concerns the asymptotic behavior as \(t \to \infty\) of viscosity solutions of the degenerate diffusion equation with the infinity-Laplacian \(u_t=\Delta_\infty u\), where \[ \Delta_\infty u= \sum_{i,j}\frac {\partial u}{\partial x_i}\frac {\partial u}{\partial x_j}\frac {\partial^2u}{\partial x_i\partial x_j}. \] Both the Cauchy problem for initial data having a compact support and the Cauchy-Dirichlet problem for bounded domains are dealt with. For the Cauchy problem and the Cauchy-Dirichlet problem under the homogeneous Dirichlet boundary condition, the optimal decay rates for bounded viscosity solutions are obtained and, in particular, it is shown that the rates are independent of the dimension of the Euclidean space. For the Cauchy-Dirichlet problem under the inhomogeneous Dirichlet boundary condition, it is shown that the solution converges to the unique stationary solution as \(t \to \infty\). The authors use the comparison principle and several kinds of barriers. For the comparison principle and the existence theorem for viscosity solutions, see \textit{G. Akagi} and \textit{K. Suzuki} [Calc. Var. Partial Differ. Equ. 31, No. 4, 457--471 (2008; Zbl 1139.35064)].
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35D40
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 5533131
zbMATH Keywords: degenerate diffusion
zbMATH Keywords: decay rate
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Cauchy-Dirichlet problem
zbMATH Keywords: comparison principle
MaRDI profile type: Publication
full work available at URL: s00208-008-0297-1
OpenAlex ID: W2063532019
Wikidata QID: Q109406481
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DOI: 10.1007/S00208-008-0297-1
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description: scientific article; zbMATH DE number 5533131