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

Flavia Smarrazzo | Zendy; Alberto Tesei | Zendy

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

Author(s) -

Flavia Smarrazzo,

Alberto Tesei

Publication year - 2012

Publication title -

networks and heterogeneous media

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.732

H-Index - 34

eISSN - 1556-181X

pISSN - 1556-1801

DOI - 10.3934/nhm.2012.7.941

Subject(s) - mathematics , entropy (arrow of time) , parabolic partial differential equation , dimension (graph theory) , cauchy problem , mathematical analysis , initial value problem , phase space , cauchy distribution , pure mathematics , physics , partial differential equation , thermodynamics

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.

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