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

Flavia Smarrazzo; Alberto Tesei; Flavia Smarrazzo; Alberto Tesei

doi:10.3934/nhm.2012.7.941

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 941-966. doi: 10.3934/nhm.2012.7.941

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

Flavia Smarrazzo ^{1
,},
Alberto Tesei ^{1
,}

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

Received: 01 March 2012 Revised: 01 October 2012
Primary: 35K55, 35R25, 35Bxx; Secondary: 34C55.

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.
- Forward-backward parabolic equations,
- entropy inequality,
- Devonshire free energy,
- phase transitions,
- interfaces
Citation: Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equationswith Devonshire free energy[J]. Networks and Heterogeneous Media, 2012, 7(4): 941-966. doi: 10.3934/nhm.2012.7.941

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Abstract

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.

References

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