#### Vol. 6, No. 7, 2013

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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

with measure-valued initial data, assuming that the regularizing term $\psi$ has logarithmic growth (the case of power-type $\psi$ was dealt with in an earlier work). We prove that this case is intermediate between the case of power-type $\psi$ and that of bounded $\psi$, 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 $\psi$), although the singular part itself need not be constant (as in the case of bounded $\psi$, 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