A free boundary characterization of measure-valued solutions for forward-backward diffusion

Abstract

We consider a nonlinear one-dimensional scalar equation of diffusion type in which, depending on the gradient of the solution, the diffusion coefficient may be positive or negative. We compare two concepts of Young measure solutions which are based on different methods to construct approximate solutions, the SP-solutions (singular perturbation) and the EM-solution (energy minimization). We show that the SP-solution can recover classical solutions where the EM-solution fails to do so, and that EM-solutions are more stable under perturbations of the initial values. We characterize the EM-solution with a free boundary problem and determine its long-time behavior.