On a class of forward-backward parabolic equations: Formation of singularities

We study the formation of singularities for the problem{u(t) = [phi(u)](xx) + epsilon[psi(u)](txx) in 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},where epsilon and Tare positive constants, Omega a bounded interval, u(0) a nonnegative Radon measure on Omega, phi a nonmonotone and nonnegative function with phi(0) = phi(infinity) = 0, and psi an increasing bounded function. We show that if u(0) is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures. (c) 2020 Elsevier Inc. All rights reserved.