Vol. 6, No. 7, 2013

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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
Abstract

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.

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