Vol. 6, No. 7, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

Michiel Bertsch, Flavia Smarrazzo and Alberto Tesei

Vol. 6 (2013), No. 7, 1719–1754

We study the initial-boundary value problem

ut = Δφ(u) + εΔ[ψ(u)]t inQ := Ω × (0,T], φ(u) + ε[ψ(u)]t = 0  in Ω × (0,T], u = u0 0 inΩ ×{0},

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.

forward-backward parabolic equations, pseudoparabolic regularization, bounded radon measures, entropy inequalities
Mathematical Subject Classification 2010
Primary: 35D99, 35K55, 35R25
Secondary: 28A33, 28A50
Received: 18 July 2012
Revised: 12 November 2012
Accepted: 20 December 2012
Published: 27 December 2013
Michiel Bertsch
Consiglio Nazionale delle Ricerche
Istituto per le Applicazioni del Calcolo “Mauro Picone”
Viale del Policlinico, 137
I-00161 Roma
Università di Roma “Tor Vergata”
Flavia Smarrazzo
Dipartimento di Matematica “G. Castelnuovo”
Universita “Sapienza” di Roma
P.le A. Moro 5
I-00185 Roma
Alberto Tesei
Dipartimento di Matematica “G. Castelnuovo”
Universita “Sapienza” di Roma
P.le A. Moro 5
I-00185 Roma