On a pseudoparabolic regularization of a forward–backward–forward equation

We consider an initial–boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward–backward–forward parabolic equation, with a bounded nonlinearity which is increasing at infinity. We prove existence of suitably defined nonnegative solutions of the problem in a space of Radon measures. Solutions satisfy several monotonicity and regularization properties; in particular, their singular part is nonincreasing and may disappear in finite time. The problem is of intrinsic mathematical interest, but also arises naturally when studying, by time reversal, the spontaneous appearance of singularities in a specific application.