Some recent results concerning a class of forward-backward parabolic equations
type: Item
instance of: scholarly article
title: Some recent results concerning a class of forward-backward parabolic equations
zbMATH Open document ID: 1222.35111
DOI: 10.4171/RLM/594
author: Flavia Smarrazzo
author: Alberto Tesei
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2011-08-02T00:00:00Z
review text: Summary: We study a quasilinear parabolic equation of forward-backward type in one space dimension, under assumptions on the nonlinearity satisfied by important mathematical models (e.g., the one-dimensional Perona-Malik equation). We first address a degenerate pseudoparabolic regularization of the equation, which takes time delay effects into account, proving existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures, as well as qualitative properties of such solutions. Then we address the vanishing viscosity limit of the regularized problem, proving that the limiting points (in a suitable topology) of the family of solutions of the regularized problem can be regarded as solutions of the original problem. In particular, we characterize the disintegration of the narrow limit of the family of Young measures associated with the absolutely continuous part of the solutions of the regularized problem, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity \(\varphi \). In the above study, the existence of a family of infinitely many entropy inequalities plays an important role.
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 28A50
zbMATH DE Number: 5935222
zbMATH Keywords: degenerate pseudoparabolic regularization
zbMATH Keywords: bounded Radon measures
zbMATH Keywords: Young measures
zbMATH Keywords: entropy inequalities
zbMATH Keywords: one space dimension
zbMATH Keywords: Perona-Malik equation
zbMATH Keywords: time delay effects
zbMATH Keywords: vanishing viscosity limit
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/rlm/594
OpenAlex ID: W2003731380
description: scientific article; zbMATH DE number 5935222