Two-phase Entropy Solutions of a Forward–Backward Parabolic Equation

This article deals with the Cauchy problem for a forward–backward parabolic equation, which is of interest in physical and biological models. Considering such an equation as the singular limit of an appropriate pseudoparabolic third-order regularization, we consider the framework of entropy solutions, namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation. Moreover, we restrict the attention to two-phase solutions, that is solutions taking values in the intervals where the parabolic equation iswell-posed, proving existence and uniqueness of such solutions.

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