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

Abstract

We study the initial-boundary value problem

$$\left\{\begin{array}{cc}{u}_t=\Delta \phi \left(u\right)+\epsilon \Delta {\left[\psi \left(u\right)\right]}_t\hfill & \text{ in}Q:=\Omega \times \left(0,T\right],\hfill \\ \phi \left(u\right)+\epsilon {\left[\psi \left(u\right)\right]}_t=0\hfill & \text{ in }\partial \Omega \times \left(0,T\right],\hfill \\ u=u_0\ge 0\text{ in}\Omega \times \left\{0\right\},\hfill \end{array}\right.$$

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.