On a class of forward–backward parabolic equations: Existence of solutions

We study the initial-boundary value problemu(t) = [phi(u)](xx) + is an element of[psi(u)](txx) in Omega x (0, T) phi(u) + is an element of[psi(u)] t = 0 in partial derivative Omega x (0, T) u = u(0) in Omega x 0,where Omega is an interval and u0 is a nonnegative Radon measure on Omega. The map phi is increasing in (0, alpha) and decreasing in (alpha, infinity) for some alpha > 0, and satisfies phi(0) = phi(infinity) = 0. The regularizing map psi is increasing and bounded. We prove existence of suitably defined nonnegative Radon measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become measure-valued after finite time. (C) 2017 Elsevier Ltd. All rights reserved.