Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 7 (2013), 1719-1754.

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

Michiel Bertsch, Flavia Smarrazzo, and Alberto Tesei

Full-text: Open access

Abstract

We study the initial-boundary value problem

u t = Δ φ ( u ) + ε Δ [ ψ ( u ) ] t  in Q : = Ω × ( 0 , T ] , φ ( u ) + ε [ ψ ( u ) ] t = 0  in  Ω × ( 0 , T ] , u = u 0 0  in Ω × { 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.

Article information

Source
Anal. PDE, Volume 6, Number 7 (2013), 1719-1754.

Dates
Received: 18 July 2012
Revised: 12 November 2012
Accepted: 20 December 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731434

Digital Object Identifier
doi:10.2140/apde.2013.6.1719

Mathematical Reviews number (MathSciNet)
MR3148065

Zentralblatt MATH identifier
1284.35121

Subjects
Primary: 35D99: None of the above, but in this section 35K55: Nonlinear parabolic equations 35R25: Improperly posed problems
Secondary: 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx] 28A50: Integration and disintegration of measures

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

Citation

Bertsch, Michiel; Smarrazzo, Flavia; Tesei, Alberto. Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity. Anal. PDE 6 (2013), no. 7, 1719--1754. doi:10.2140/apde.2013.6.1719. https://projecteuclid.org/euclid.apde/1513731434


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