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

We study the initial-boundary value problem u_t = Δφ(u) + εΔ[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.