# American Institute of Mathematical Sciences

December  2012, 7(4): 941-966. doi: 10.3934/nhm.2012.7.941

## Entropy solutions of forward-backward parabolic equations with Devonshire free energy

 1 Department of Mathematics "G. Castelnuovo", University of Rome "La Sapienza", P.le A. Moro 5, I-00185 Rome, Italy, Italy

Received  March 2012 Revised  October 2012 Published  December 2012

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.
Citation: Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941
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