ENTROPY SOLUTIONS OF FORWARD-BACKWARD PARABOLIC EQUATIONS WITH DEVONSHIRE FREE ENERGY

A class of quasilinear parabolic equations of forward-backward type u(t) = [phi(u)](xx) in one space dimension is addressed, under assumptions on the nonlinear term phi which hold for a number of mathematical models in the theory of phase transitions. The notion of a three-phase solution to the Cauchy problem associated with the aforementioned equation is introduced. Then the time evolution of three-phase solutions is investigated, relying on a suitable entropy inequality satisfied by such a solution. In particular, it is proven that transitions between stable phases must satisfy certain admissibility conditions.