The Cauchy problem for linear growth functionals
type: Item
instance of: scholarly article
title: The Cauchy problem for linear growth functionals
zbMATH Open document ID: 1028.35054
published in: Journal of Evolution Equations
publication date: 2004-01-18T00:00:00Z
review text: The following Cauchy problem is considered \[ \frac{\partial u}{\partial t}=\text{div }\mathbf a(x,Du)\quad \text{in}\quad Q=(0,\infty)\times\mathbb R^N;\qquad u(0,x)=u_0(x)\quad \text{in}\quad x\in\mathbb R^N,\tag{*} \] where \(u_0\in L_{loc}^1(\mathbb R^N)\), \(\mathbf a(x,\xi)=\nabla_\xi f(x,\xi)\) and \(f:\mathbb R^N\times\mathbb R^N\to\mathbb R\) is a function with linear growth as \(\|\xi\|\to\infty\) satisfying some additional assumptions. An example of a function \(f(x,\xi)\) covered by these assumptions is \(f(x,\xi)=\sqrt{1+\|\xi\|^2}\) but the function \(f(x,\xi)=\|\xi\|\) does not satisfy these assumptions. Existence and uniqueness results are stated for the initial data in \(L_{loc}^1(\mathbb R^N)\).
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1921149
zbMATH Keywords: existence and uniqueness results
zbMATH Keywords: bounded variation
author: F. Andreu
author: V. Caselles
author: J. M. Mazón
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
full work available at URL: s000280300003
OpenAlex ID: W4249004157
DOI: 10.1007/S000280300003
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description: scientific article; zbMATH DE number 1921149