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This paper study the well--posedness of the entropy formulation given byPlotnikov in [{Differential Equations}, 30 (1994), pp. 614--622] forforward-backward parabolic problem obtained as singular limit of a properpseudoparabolic approximation. It was proved in [C. Mascia, A. Terracina, andA. Tesei, {Arch.\ Ration.\ Mech.\ Anal.}, 194 (2009), pp. 887--925] that suchformulation gives uniqueness when the solution takes values in the stablephases. Here we consider the situation in which unstable phase is taken inaccount, proving that, in general, uniqueness does not hold.

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