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We consider a lattice regularization for an ill-posed diffusion equation withtrilinear constitutive law and study the dynamics of phase interfaces in theparabolic scaling limit. Our main result guarantees for a certain class ofsingle-interface initial data that the lattice solutions satisfy asymptoticallya free boundary problem with hysteretic Stefan condition. The key challenge inthe proof is to control the microscopic fluctuations that are inevitablyproduced by the backward diffusion when a particle passes the spinodal region.

Hysteresis and phase transitions in a lattice regularization of an ill-posed forward-backward diffusion equation