Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 785-812.

In this paper, we show the existence and qualitative properties of traveling wave solutions to the Allen–Cahn equation with fractional Laplacians. A key ingredient is the estimation of the traveling speed of traveling wave solutions.

DOI : 10.1016/j.anihpc.2014.03.005
Classification : 35B32, 35C07, 35J20, 35R09, 35R11, 45G05, 47G10
Mots clés : Traveling wave solution, Traveling speed, Allen–Cahn equation, Fractional Laplacian, Continuation method, Hamiltonian identity
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Gui, Changfeng; Zhao, Mingfeng. Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 785-812. doi : 10.1016/j.anihpc.2014.03.005. http://www.numdam.org/articles/10.1016/j.anihpc.2014.03.005/

[1] N.D. Alikakos, N.I. Katzourakis, Heteroclinic travelling waves of gradient diffusion systems, Trans. Am. Math. Soc. 363 (2011), 1365 -1397 | MR | Zbl

[2] S.M. Allen, J.W. Cahn, A microscope theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 no. 6 (1979), 1085 -1095

[3] L. Ambrosio, X. Cabré, Entire solutions of semilinear elliptic equations in 𝐑 3 and a conjecture of De Giorgi, J. Am. Math. Soc. 13 no. 4 (2000), 725 -739 | MR | Zbl

[4] D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33 -76 | MR | Zbl

[5] G. Barles, H.M. Soner, P.E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 no. 2 (1993), 439 -469 | MR | Zbl

[6] Peter W. Bates, Xinfu Chen, Adam J.J. Chma, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var. Partial Differ. Equ. 24 no. 3 (2005), 261 -281 | MR | Zbl

[7] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal. 138 no. 2 (1997), 105 -136 | MR | Zbl

[8] J. Bebernes, D. Eberly, Mathematical problems from combustion theory, Appl. Math. Sci. vol. 83 , Springer-Verlag, New York (1989) | MR | Zbl

[9] H. Berestycki, F. Hamel, Generalized travelling waves for reaction–diffusion equations, Perspectives in Nonlinear Partial Differential Equations. In honor of H. Brezis, Contemp. Math. vol. 446 , Amer. Math. Soc. (2007), 101 -123 | Zbl

[10] H. Berestycki, F. Hamel, Reaction–Diffusion Equations and Propagation Phenomena, Applied Mathematical Sciences , Springer-Verlag (2014) | Zbl

[11] Henri Berestycki, Louis Nirenberg, Travelling fronts in cylinders, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9 no. 5 (1992), 497 -572 | EuDML | Numdam | MR | Zbl

[12] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems I: periodic framework, J. Eur. Math. Soc. 2 (2005), 173 -213 | EuDML | MR | Zbl

[13] J. Bony, P. Courrège, P. Priouret, Semi-groups de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier 18 no. 2 (1969), 369 -521 | EuDML | Numdam | MR | Zbl

[14] N.F. Britton, Reaction–Diffusion Equations and their Applications to Biology, Academic Press Inc., London (1986) | MR | Zbl

[15] A. Blumen, J. Klafter, I.M. Sokolov, Fractional kinetics, Phys. Today 55 (2002), 48 -54

[16] Xavier Cabré, Eleonora Cinti, Energy estimates and 1-d symmetry for nonlinear equations involving the half-Laplacian, Discrete Contin. Dyn. Syst. 28 no. 3 (2010), 1179 -1206 | MR | Zbl

[17] Xavier Cabré, Eleonora Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, preprint, 2011. | MR

[18] Xavier Cabré, Neus Cónsul and José Vicente Mandé, Traveling wave solutions in a halfspace for boundary reactions, preprint, 2010.

[19] X. Cabré, J.M. Roquejoffre, Propagation de fronts dans les équations de Fisher–KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. 347 (2009), 1361 -1366 | MR | Zbl

[20] X. Cabré, J.-M. Roquejoffre, The influence of fractional diffusion in Fisher–KPP equations, preprint, 2011. | MR

[21] Xavier Cabré, Yannick Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates, preprint, 2010. | Numdam | MR

[22] X. Cabré, Yannick Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, preprint, 2011. | MR

[23] Xavier Cabré, Joan Solà-Morales, Layer solutions in a half-space for boundary reactions, Commun. Pure Appl. Math. 58 no. 12 (2005), 1678 -1732 | MR | Zbl

[24] L. Caffarelli, A. Mellet, Y. Sire, Traveling waves for a boundary reaction–diffusion equation, preprint, 2011. | MR

[25] L.A. Caffarelli, J.M. Roquejoffre, O. Savin, Nonlocal minimal surfaces, Commun. Pure Appl. Math. 63 no. 9 (2010), 1111 -1144 | MR | Zbl

[26] Luis Caffarelli, Luis Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ. 32 (2007), 1245 -1260 | MR | Zbl

[27] L.A. Caffarelli, E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differ. Equ. 41 (2011), 103 -240 | MR | Zbl

[28] X. Chen, Generation and propagation of interfaces in reaction–diffusion equations, J. Differ. Equ. 96 (1992), 116 -141 | MR | Zbl

[29] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ. 2 (1997), 125 -160 | MR | Zbl

[30] X. Chen, J.S. Guo, F. Hamel, H. Ninomiya, J.M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24 (2007), 369 -393 | EuDML | Numdam | MR | Zbl

[31] A.M. Cuitiño, M. Koslowski, M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids 50 (2002), 2597 -2635 | MR | Zbl

[32] E. De Giorgi, Convergence problems for functionals and operators, Rome, 1978, E. De Giorgi, et al. (ed.) Proc. Int. Meeting on Recent Methods in Nonlinear Analysis , Pitagora, Bologna (1979) | MR | Zbl

[33] A. De Masi, T. Gobron, E. Presutti, Traveling fronts in non-local evolution equations, Arch. Ration. Mech. Anal. 132 (1995), 143 -205 | MR | Zbl

[34] D. Del-Castillo-Negrete, B.A. Carreras, V.E. Lynch, Front dynamics in reaction diffusion systems with Levy flights: a fractional diffusion approach, Phys. Rev. Lett. 91 (2003)

[35] D. Del-Castillo-Negrete, Truncation effects in superdiffusive front propagation with Levy flights, Phys. Rev. E 79 (2009)

[36] M. Del Pino, M. Kowalczyk, J. Wei, On De Giorgi's in dimension N9 , Ann. Math. (2) 174 no. 3 (2011), 1485 -1569 | MR | Zbl

[37] Lawrence C. Evans, Partial Differential Equations, American Mathematical Society (1998) | MR | Zbl

[38] L.C. Evans, H.M. Soner, P.E. Souganidis, Phase and transitions and generalized motion by mean curvature, Commun. Pure Appl. Math. 45 (1992), 1097 -1123 | Zbl

[39] P.C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS–NSF Regional Conference, Series in Applied Mathematics vol. 53 (1988) | MR | Zbl

[40] P.C. Fife, J.B. Mcleod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal. 65 (1977), 335 -361 | MR | Zbl

[41] R. Fisher, The wave of advance of advantageous genes, Annu. Eugen. 7 (1937), 355 -369 | JFM

[42] Rupert L. Frank, Enno Lenzmann, Uniqueness and nondegeneracy of ground states for (-Δ) s q+q-q α+1 =0 in R , arXiv:1009.4042 (2010)

[43] A. Garroni, S. Müller, Γ-limit of a phase-field model of dislocations, SIAM J. Math. Anal. 36 (2005), 1943 -1964 | MR | Zbl

[44] A. Garroni, S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal. 181 no. 3 (2006), 535 -578 | MR | Zbl

[45] N. Ghoussoub, C. Gui, On a conjecture of de Giorgi and some related problems, Math. Ann. 311 (1998), 481 -491 | MR | Zbl

[46] N. Ghoussoub, C. Gui, On de Giorgi's conjecture in dimensions 4 and 5, Ann. Math. 157 (2003), 313 -334 | MR | Zbl

[47] David Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (2001) | MR | Zbl

[48] B.H. Gilding, R. Kersner, Travelling Waves in Nonlinear Diffusion–Convection Reaction, Prog. Nonlinear Differ. Equ. Appl. vol. 60 , Birkhäuser Verlag, Basel (2004) | MR | Zbl

[49] A.A. Golovin, Y. Nec, A.A. Nepomnyashchy, Front-type solutions of fractional Allen–Cahn equation, Physica D 237 (2008), 3237 -3251 | MR | Zbl

[50] M.D.M. Gonzalez, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differ. Equ. 36 no. 2 (2009), 173 -210 | MR | Zbl

[51] C. Gui, Symmetry of traveling wave solutions to the Allen–Cahn equation in 𝐑 2 , Arch. Ration. Mech. Anal. 203 no. 3 (2012), 1037 -1065 | MR | Zbl

[52] C. Gui, Properties of traveling wave solutions to Allen–Cahn equation in all dimensions, preprint.

[53] C. Gui, T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, to appear in Calc. Var. Partial Differ. Equ. | MR

[54] C. Gui, M. Zhao, Asymptotic formula for the speed of traveling wave solutions to Allen–Cahn equations, preprint.

[55] F. Hamel, R. Monneau, Solutions of semilinear elliptic equations in 𝐑 n with conical shaped level sets, Commun. Partial Differ. Equ. 25 (2000), 769 -819 | MR | Zbl

[56] F. Hamel, R. Monneau, J.M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Éc. Norm. Super. 37 (2004), 469 -506 | EuDML | Numdam | MR | Zbl

[57] F. Hamel, R. Monneau, J.M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst. 13 (2005), 1069 -1096 | MR | Zbl

[58] F. Hamel, R. Monneau, J.M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst. 14 (2006), 75 -92 | MR | Zbl

[59] F. Hamel, N. Nadirashvili, Traveling waves and entire solutions of the Fisher–KPP equation in 𝐑 n , Arch. Ration. Mech. Anal. 157 (2001), 91 -163 | MR | Zbl

[60] F. Hamel, L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differ. Equ. 249 (2010), 1726 -1745 | MR | Zbl

[61] D. Hernandez, R. Barrio, C. Varea, Wave-front dynamics in systems with directional anomalous diffusion, Phys. Rev. E 74 (2006)

[62] Cyril Imbert, Panagiotis E. Souganidis, Phasefield theory for fractional diffusion–reaction equations and applications, preprint.

[63] Y. Kim, K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymetric positive kernels, preprint, 2011. | MR

[64] Y. Kim, K. Lee, Regularity results for fully nonlinear parabolic integro-differential operators, preprint, 2011. | MR

[65] J. Klafter, R. Metzler, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1 -77 | MR | Zbl

[66] A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, Etude de l' équation de diffusion avec accroissement de la quantité de matiére, et son application á un probléme biologique, Vestn. Mosk. Univ. 17 (1937), 1 -26

[67] N.S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180 , Springer-Verlag, New York (1972) | MR | Zbl

[68] C.D. Levermore, J.X. Xin, Multidimensional stability of traveling waves in a bistable reaction diffusion equation II, Commun. Partial Differ. Equ. 17 (1992), 1901 -1924 | MR | Zbl

[69] R. Mancinelli, D. Vergni, A. Vulpiani, Front propagation in reactive systems with anomalous diffusion, Physica D 185 (2003), 175 -195 | MR | Zbl

[70] R. Mancinelli, D. Vergni, A. Vulpiani, Superfast front propagation in reactive systems with non-Gaussian diffusion, Europhys. Lett. 60 (2002), 532 -538 | Zbl

[71] Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire, Existence and asymptotics of fronts in non-local combustion models, preprint, 2011. | MR

[72] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 no. 31 (2004), 161 -208 | MR | Zbl

[73] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal. 98 no. 2 (1987), 123 -142 | MR | Zbl

[74] C.B. Muratov, F. Posta, S.Y. Shvartsman, Autocrine signal transmission with extracellular ligand degradation, Phys. Biol. 6 no. 1 (2009), 016006

[75] Y. Necb, A.A. Nepomnyashchy, V.A. Volpert, Exact solutions in front propagation problems with superdiffusion, Physica D 239 (2010), 134 -144 | MR | Zbl

[76] H. Ninomiya, M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen–Cahn equations, J. Differ. Equ. 213 (2005), 204 -233 | MR | Zbl

[77] H. Ninomiya, M. Taniguchi, Global stability of traveling curved fronts in the Allen–Cahn equations, Discrete Contin. Dyn. Syst. 15 (2006), 819 -832 | MR | Zbl

[78] G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, preprint, 2010 (available from arXiv). | MR

[79] O. Savin, Regularity of level sets in phase transitions, Ann. Math. 169 no. 1 (2009), 41 -78 | MR | Zbl

[80] O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, preprint, 2010. | MR

[81] O. Savin, E. Valdinoci, Γ-convergence for nonlocal phase transitions, preprint, 2011. | Numdam | MR

[82] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, preprint, 2011.

[83] L. Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, preprint, 2012. | MR

[84] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen–Cahn equation, SIAM J. Math. Anal. 39 (2007), 319 -344 | MR | Zbl

[85] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations, J. Differ. Equ. 246 (2009), 2103 -2130 | MR | Zbl

[86] A.I. Volpert, V.A. Volpert, V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, AMS, Providence (1994) | MR | Zbl

[87] V. Volpert, A. Volpert, Existence of multidimensional travelling waves in the bistable case, C. R. Acad. Sci., Ser. 1 Math. 328 no. 3 (1999), 245 | MR | Zbl

[88] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differ. Equ. 183 (2002), 434 -461 | MR | Zbl

[89] J. Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), 161 -230 | MR | Zbl

[90] J.X. Xin, Multidimensional stability of traveling waves in a bistable reaction diffusion equation I, Commun. Partial Differ. Equ. 17 (1992), 1889 -1899 | MR | Zbl

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