Measure-valued solutions of nonlinear parabolic equations with logarithmic diffusion
type: Item
instance of: scholarly article
title: Measure-valued solutions of nonlinear parabolic equations with logarithmic diffusion
zbMATH Open document ID: 1329.35176
author: Luigi Orsina
author: Maria Michaela Porzio
author: Flavia Smarrazzo
published in: Journal of Evolution Equations
publication date: 2015-10-20T00:00:00Z
review text: The authors study the existence and regularity of a Radon measure-valued solution for a class of Cauchy problems for nonlinear degenerate parabolic equations of the form \[ \begin{cases} u_t=\Delta (\psi(u)) & \text{in} \;\Omega\times(0,T),\\ u=0 & \text{on} \;\partial\Omega\times(0,T),\\ u=u_0 & \text{in} \;\Omega\times\{0\}, \end{cases} \] where \(\Omega\subset \mathbb{R}^N,\) \(N\geq2,\) is a bounded and smooth domain and the initial datum \(u_0\) is a bounded Radon measure. Regarding the function \(\psi\in C^\infty(\mathbb{R}),\) it is supposed that \(\psi(0)=0,\) \(\psi'(s)>0\) and there exist \(s_0>0\) and \(\beta>\alpha>0\) such that \[ \dfrac{\alpha}{1+|s|}\leq \psi'(s)\leq \dfrac{\beta}{1+|s|}\quad \text{for all } |s|\geq s_0. \] In particular, it is proved that a regularizing effect appears if the initial datum is diffused with respect to the ``\(C_2\)-capacity'', since in this case the solution becomes a summable function. Moreover, the uniqueness of measure-valued solutions is studied as well.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35R06
zbMATH DE Number: 6496639
zbMATH Keywords: Radon measures
zbMATH Keywords: nonlinear degenerate parabolic equations
zbMATH Keywords: existence and uniqueness results
reviewed by: Dian K. Palagachev
MaRDI profile type: Publication
full work available at URL: s00028-015-0275-5
OpenAlex ID: W2080323458
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DOI: 10.1007/S00028-015-0275-5
description: scientific article; zbMATH DE number 6496639