PSEUDOPARABOLIC REGULARIZATION OF FORWARD-BACKWARD PARABOLIC EQUATIONS: A LOGARITHMIC NONLINEARITY

We study the initial-boundary value problem {u(t) = Delta phi(u) + epsilon Delta[psi(u)](t) in Q := Omega x (0, T], phi(u) + epsilon[psi(u)](t) = 0 in partial derivative Omega x (0, T], u = u(0) >= 0 in Omega x {0}, 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.