Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation
type: Item
instance of: scholarly article
title: Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation
zbMATH Open document ID: 1334.35107
author: Claudete M. Webler
author: Sergio Segura de León
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2015-10-15T00:00:00Z
review text: In the very interesting paper under review, the authors introduce a new approach to the Dirichlet problem for the total variation flow in a bounded domain and analyze the associated inhomogeneous problem. Precisely, the paper deals with the problem \[ \begin{cases} \dfrac{\partial u}{\partial t}- \operatorname{div} \left(\dfrac{Du}{|Du|}\right)=f(x,t) & \text{in } \Omega\times(0,+\infty),\\ u=0 & \text{on } \partial\Omega\times(0,+\infty),\\ u(x,0)=u_0(x) & \text{in } \Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with Lipschitz continuous boundary, \(f\in L^1_{\text{loc}}(0,+\infty;L^2(\Omega))\), and \(u_0\in L^2(\Omega)\). The main result of the paper asserts the existence of a unique solution \(u\in L^1_{\text{loc}}(0,+\infty;BV(\Omega))\cap C([0,+\infty);L^2(\Omega))\) to the above problem. The authors compare, moreover, the solutions corresponding to different data and study the long-term behaviour of the solutions. Explicit examples of radial solutions are shown as well.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K67
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6494623
zbMATH Keywords: inhomogeneous evolution equations
zbMATH Keywords: singular diffusion equations
zbMATH Keywords: \(1\)-Laplacian
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: total variation flow
zbMATH Keywords: \(L^2\)initial data
zbMATH Keywords: \(p\)-Laplacian
reviewed by: Dian K. Palagachev
MaRDI profile type: Publication
full work available at URL: s00030-015-0320-7
OpenAlex ID: W1976949322
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DOI: 10.1007/S00030-015-0320-7
description: scientific article; zbMATH DE number 6494623