Strong solutions in \(L^ 1\) of degenerate parabolic equations
type: Item
instance of: scholarly article
title: Strong solutions in \(L^ 1\) of degenerate parabolic equations
zbMATH Open document ID: 0828.35050
author: Ronald F. Gariepy
author: Philippe Benilan
published in: Journal of Differential Equations
publication date: 1996-01-14T00:00:00Z
review text: The paper deals with the degenerate parabolic problem of the form \(u_t = \Delta \varphi (u) + \text{div} F(u)\) on \(Q = (0,T) \times \Omega\), \(u(0, \cdot) = u_0\) on \(\Omega\), where \(\Omega \subset \mathbb{R}^N\), \(\varphi \in C^1 (\mathbb{R})\), \(F \in C^1 (\mathbb{R})^N\), \(u_0 \in L^\infty (\mathbb{R}^N)\). Assuming that \(\varphi' > 0\) a.e. on \(\mathbb{R}\) and \(|F' |^2 \leq \sigma \varphi'\) for some \(\sigma \in C (\mathbb{R})\), the authors prove that the above problem has a strong solution \(u\), i.e. \(u_t\), \(\Delta \varphi (u)\), \(\text{div} F(u)\) are functions in \(L^1_{\text{loc}} (Q)\). The proof is based on the theory of \(BV\) functions in several variables and geometric measures.
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 784949
zbMATH Keywords: \(BV\) functions in several variables
zbMATH Keywords: geometric measures
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1099
OpenAlex ID: W2008140516
DOI: 10.1006/JDEQ.1995.1099
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description: scientific article; zbMATH DE number 784949