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- Publisher
- Springer Journals
- Copyright
- Copyright © Springer International Publishing AG 2017
- Subject
- Mathematics; Analysis
- ISSN
- 1021-9722
- eISSN
- 1420-9004
- DOI
- 10.1007/s00030-017-0468-4
- Publisher site
- See Article on Publisher Site
Abstract
The purpose of this paper is to investigate the time behavior of the solution of a weighted p-Laplacian evolution equation, given by 0.1ut=divγ|∇u|p-2∇uon(0,∞)×S,γ|∇u|p-2∇u·η=0on(0,∞)×∂S,u(0,·)=u0onS,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t} = \text {div} \left( \gamma |\nabla u|^{p-2}\nabla u \right) &{} \text {on } (0,\infty )\times S, \\ \gamma |\nabla u|^{p-2}\nabla u\cdot \eta =0 &{} \text {on } (0,\infty )\times \partial S, \\ u(0,\cdot )=u_{0} &{} \text {on } S,\end{array}\right. } \end{aligned}$$\end{document}where n∈N\{1}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \in \mathbb {N}{\setminus } \{1\}$$\end{document}, p∈(1,∞)\{2}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \in (1,\infty ){\setminus } \{2\}$$\end{document}, S⊆Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S\subseteq \mathbb {R}^{n}$$\end{document} is an open, bounded and connected set of class C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{1}$$\end{document}, η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta $$\end{document} is the unit outer normal on ∂S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial S$$\end{document}, and γ:S→(0,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma :S\rightarrow (0,\infty )$$\end{document} is a bounded function which can be extended to an Ap\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{p}$$\end{document}-Muckenhoupt weight on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^{n}$$\end{document}. It will be proven that the solution of (0.1) converges in L1(S)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{1}(S)$$\end{document} to the average of the initial value u0∈L1(S)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u_{0} \in L^{1}(S)$$\end{document}. Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.
Journal
Nonlinear Differential Equations and Applications NoDEA – Springer Journals
Published: Aug 1, 2017
Keywords: Weighted p-Laplacian evolution equation; Asymptotic results for nonlinear semigroups; Nonlocal diffusion; Neumann boundary conditions; Primary 35B40; Secondary 47H20