DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

doi:10.1016/S0252-9602(17)30104-2

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (6): 1740-1760.doi: 10.1016/S0252-9602(17)30104-2

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DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

Tianyuan XU¹, Chunhua JIN¹, Shanming JI²

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematics, South China University of Technology, Guangzhou 510641, China

Received:2016-06-28 Revised:2017-06-16 Online:2017-12-25 Published:2017-12-25
Contact: Shanming JI E-mail:jishanming7@126.com
Supported by:
This work was supported by the Program for New Century Excellent Talents in University of the Ministry of Education (NCET-13-0804), NSFC (11471127, 11371153), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029), the Excellent Young Teachers Program of Guangdong Province, and Special support program of Guangdong Province. The research of Xu was supported by the Innovation Project of Graduate School of South China Normal University (2016lkxm70), the High-Level University Developments Innovation Project of Graduate School of South China Normal University (2016YN05) and the Top-Notch Graduate Foundation of South China Normal University (2016028). The research of Ji was supported by the Fundamental Research Funds for the Central Universities (2017BQ109) and China Postdoctoral Science Foundation (2017M610517).

Abstract

Tianyuan XU, Chunhua JIN, Shanming JI. DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL[J].Acta mathematica scientia,Series B, 2017, 37(6): 1740-1760.

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[1] Allen S M, Cahn J W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall, 1979, 27:1085-1095
[2] Barbera E, Currò C, Valenti G. On discontinuous travelling wave solutions for a class of hyperbolic reactiondiffusion models. Physica D, 2015, 308:116-126
[3] Bertsch M, Passo R D. Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch Rat Mech Anal, 1992, 117:349-387
[4] Cahn J W, Hilliard J E. Free energy of a nonuniform system, I:interfacial free energy. J Chem Phys, 1958, 28(2):258-267
[5] Chmaj A, Ren X. Multiple layered solutions of the nonlocal bistable equation. Physica D, 2000, 147:135-154
[6] Dickman R. Discontinuous phase transition in a dimer lattice gas. J Chem Phys, 2012, 136:4678-4686
[7] Galenko P K, Sanches F I, Elder K R. Traveling wave profiles for a crystalline front invading liquid states:analytical and numerical solutions. Physica D, 2015, 308:1-10
[8] Garrione M, Sanchez L. Monotone traveling waves for reaction-diffusion equations involving the curvature operator. Bound Value Probl, 2015, 2015:1-31
[9] Holden H, Karlsen K H, Mitrovic D, Panov E Y. Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function. Acta Math Sci, 2009, 29B(6):1573-1612
[10] Jordan P M. Growth and decay of shock and acceleration waves in a traffic flow model with relaxation. Physica D, 2005, 207:220-229
[11] Kruzhkov S N. First order quasilinear equations in several independent variables. Mat Sb, 1970, 81:228-255(English transl Math USSR-Sb, 1970, 10:217-243)
[12] Landman K A, Pettet G J, Newgreen D F. Chemotactic cellular migration:smooth and discontinuous travelling wave solutions. SIAM J Appl Math, 2003, 63(5):1666-1681
[13] Li M, Geng H, Long F et al. Discontinuous structural phase transition of Sn-Bi melts. J Mol Liq, 2015, 204:27-32
[14] Liu H X, Pan T. Pointwise convergence rate of vanishing viscosity approximations for scalar conservation laws with boundary. Acta Math Sci, 2009, 29B(1):111-128
[15] Mai D T. Traveling waves of an elliptic-hyperbolic model of phase transitions via varying viscositycapillarity. J Differ Equ, 2011, 251:439-456
[16] Pang P, Wang Y H, Yin J X. Free-boundary problem for a singular diffusion equation. J Math Anal Appl, 2002, 265:414-429
[17] Passo R D. Uniqueness of the entropy solution of a strongly degenerate parabolic equation. Commun Partial Differ Equ, 1993, 18:265-279
[18] Rosenau P. Free-energy functionals at the high-gradient limit. Phys Rev A, 1990, 41:2227-2230
[19] Tacan F, Bekir A. Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl Math Comput, 2009, 207:279-282
[20] Wang C, Yin J, Wang Z. Similar entropy solutions of a singular diffusion equation. Comput Math Appl, 2005, 49:1059-1068
[21] Wang C, Yin J, Lu B. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete Contin Dyn Syst Ser B, 2010, 3:1211-1236
[22] Wu Z Q, Yin J X. Some properties of functions in BV_x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeastern Math J, 1989, 5:395-422
[23] Yin J X, Jin C H. Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources. Proc R Soc Edinb Sect A-Math, 2009, 139:1179-1207
[24] Yin J X, Li H L, Pang P Y H. BV solutions of a singular diffusion equation. Math Nachr, 2003, 253:92-106

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

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