Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

with measure-valued initial data, assuming that the regularizing term $ψ$ has logarithmic growth (the case of power-type $ψ$ was dealt with in an earlier work). We prove that this case is intermediate between the case of power-type $ψ$ and that of bounded $ψ$ , to be addressed in a forthcoming paper. Specifically, the support of the singular part of the solution with respect to the Lebesgue measure remains constant in time (as in the case of power-type $ψ$ ), although the singular part itself need not be constant (as in the case of bounded $ψ$ , where the support of the singular part can also increase). However, it turns out that the concentrated part of the solution with respect to the Newtonian capacity remains constant.

Keywords

forward-backward parabolic equations, pseudoparabolic regularization, bounded radon measures, entropy inequalities

Mathematical Subject Classification 2010

Primary: 35D99, 35K55, 35R25

Secondary: 28A33, 28A50

Milestones

Received: 18 July 2012

Revised: 12 November 2012

Accepted: 20 December 2012

Published: 27 December 2013

Authors

Michiel Bertsch
Consiglio Nazionale delle Ricerche Istituto per le Applicazioni del Calcolo “Mauro Picone” Viale del Policlinico, 137 I-00161 Roma Italy	Università di Roma “Tor Vergata” Roma Italy

Flavia Smarrazzo
Dipartimento di Matematica “G. Castelnuovo” Universita “Sapienza” di Roma P.le A. Moro 5 I-00185 Roma Italy

Alberto Tesei
Dipartimento di Matematica “G. Castelnuovo” Universita “Sapienza” di Roma P.le A. Moro 5 I-00185 Roma Italy

Vol. 6, No. 7, 2013

Michiel Bertsch, Flavia Smarrazzo and Alberto Tesei

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors