Non-uniqueness results for entropy two-phase solutions of forward–backward parabolic problems with unstable phase

This paper studies the well-posedness of the entropy formulation given by Plotnikov (1994) in [23] for forward–backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in Mascia et al. (2009) [20] that such a formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold.