# 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 and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941
##### References:
 [1] G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439. doi: 10.1137/0524082. [2] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [3] L. T. T. Bui, F. Smarrazzo, and A. Tesei, forthcoming. [4] L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620. doi: 10.1142/S0218202504003763. [5] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1991. [6] C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in: "Asymptotic Analysis and Singularities" (edited by H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida), 451-478, Advanced Studies in Pure Mathematics, 47-2 (Math. Soc. Japan, 2007). [7] C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Rational Mech. Anal., 194 (2009), 887-925. doi: 10.1007/s00205-008-0185-6. [8] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. doi: 10.2307/2001511. [9] V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), 457-486. doi: 10.1080/03605309808821353. [10] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. [11] P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Equ., 30 (1994), 614-622. [12] D. Serre, "Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves," Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374. [13] F. Smarrazzo and A. Terracina, Local existence and uniqueness of two-phase entropy solutions, Discrete Contin. Dyn. Syst. (to appear). [14] A. Terracina, Qualitative behaviour of the two-phase entropy solution of a forward-backward parabolic equation, SIAM J. Math. Anal., 43 (2011), 228-252. doi: 10.1137/090778833. [15] M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1995), 349-394.

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##### References:
 [1] G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439. doi: 10.1137/0524082. [2] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [3] L. T. T. Bui, F. Smarrazzo, and A. Tesei, forthcoming. [4] L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620. doi: 10.1142/S0218202504003763. [5] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1991. [6] C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in: "Asymptotic Analysis and Singularities" (edited by H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida), 451-478, Advanced Studies in Pure Mathematics, 47-2 (Math. Soc. Japan, 2007). [7] C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Rational Mech. Anal., 194 (2009), 887-925. doi: 10.1007/s00205-008-0185-6. [8] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. doi: 10.2307/2001511. [9] V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), 457-486. doi: 10.1080/03605309808821353. [10] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. [11] P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Equ., 30 (1994), 614-622. [12] D. Serre, "Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves," Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374. [13] F. Smarrazzo and A. Terracina, Local existence and uniqueness of two-phase entropy solutions, Discrete Contin. Dyn. Syst. (to appear). [14] A. Terracina, Qualitative behaviour of the two-phase entropy solution of a forward-backward parabolic equation, SIAM J. Math. Anal., 43 (2011), 228-252. doi: 10.1137/090778833. [15] M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1995), 349-394.
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