March  2012, 32(3): 1011-1046. doi: 10.3934/dcds.2012.32.1011

Multi-dimensional traveling fronts in bistable reaction-diffusion equations

1. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama 2-12-1-W8-38, Tokyo 152-8552, Japan

Received  October 2010 Revised  September 2011 Published  October 2011

This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.
Citation: Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1084-1095. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, Partial Differential Equations and Related Topics, ed. J. A. Goldstein, Lecture Notes in Mathematics, 446 (1975) 5-49.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$, Indiana Univ. Math. J. 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[5]

J. Buckmaster, Polyhedral flames--an exercise in bimodal bifurcation analysis, SIAM J. Appl. Math., 44 (1984), 40-55. doi: 10.1137/0144005.  Google Scholar

[6]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[7]

X. Chen, J-S. Guo, F. Hamel, H. Ninomiya and J-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. I. H. Poincaré, AN 24 (2007), 369-393.  Google Scholar

[8]

M. del Pino, M. Kowalczyk and J. Wei, On de Giorgi conjecture in dimension $N\geq9$,, Annals of Math. (to appear)., ().   Google Scholar

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.  Google Scholar

[10]

R. A. Fisher, The advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.  Google Scholar

[12]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Scient. Ec. Norm. Sup. 4ème série, t.37 (2004), 469-506.  Google Scholar

[13]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.  Google Scholar

[14]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.  Google Scholar

[15]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rat. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.  Google Scholar

[16]

F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101.  Google Scholar

[17]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, AN 23 (2006), 283-329.  Google Scholar

[18]

Y. I. Kanel', Certain problems on equations in the theory of burning, Soviet. Math. Dokl., 2 (1961), 48-51.  Google Scholar

[19]

Y. I. Kanel', Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.), 59 (1962), 245-288.  Google Scholar

[20]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[21]

K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems, Phys. A, 116 (1982), 573-593. doi: 10.1016/0378-4371(82)90178-9.  Google Scholar

[22]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.  Google Scholar

[23]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II, Comm. Par. Diff. Eq., 17 (1992), 1901-1924.  Google Scholar

[24]

H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Par. Diff. Eq. , 34 (2009), 976-1002.  Google Scholar

[25]

J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines, IEEE Trans. Circuit Theory, CT-12 (1965), 400-412. Google Scholar

[26]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: Theory and applications I, GAKUTO Internat. Ser. Math. Sci. Appl., 13 (2000), 206-221.  Google Scholar

[27]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.  Google Scholar

[28]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.  Google Scholar

[29]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbfR^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651.  Google Scholar

[30]

V. Pérez-Muñuzuri, M. Gómez-Gesteira, A. P. Muñuzuri, V. A. Davydov and V. Pérez-Villar, V-shaped stable nonspiral patterns, Physical Review E, 51 (1995), 845-847. Google Scholar

[31]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, Berlin, 1984.  Google Scholar

[32]

J-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Annali di Matematica, 188 (2009), 207-233.  Google Scholar

[33]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.  Google Scholar

[34]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[35]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  Google Scholar

[36]

F. A. Smith and S. F. Pickering, Bunsen flames of unusual structure, Proceedings of the Symposium on Combustion, Vol. 1-2 (1948), 24-26. doi: 10.1016/S1062-2888(65)80006-5.  Google Scholar

[37]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.  Google Scholar

[38]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[39]

M. Taniguchi, Traveling fronts in perturbed multistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. - Supplement 2011 (The proceedings for the 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications), (to appear). Google Scholar

[40]

J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237. doi: 10.1063/1.440418.  Google Scholar

[41]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Comm. Par. Diff. Eq., 17 (1992), 1889-1899.  Google Scholar

[42]

H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation, J. Dynam. Differential Equations, 13 (2001), 323-353. doi: 10.1023/A:1016632124792.  Google Scholar

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1084-1095. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, Partial Differential Equations and Related Topics, ed. J. A. Goldstein, Lecture Notes in Mathematics, 446 (1975) 5-49.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$, Indiana Univ. Math. J. 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[5]

J. Buckmaster, Polyhedral flames--an exercise in bimodal bifurcation analysis, SIAM J. Appl. Math., 44 (1984), 40-55. doi: 10.1137/0144005.  Google Scholar

[6]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[7]

X. Chen, J-S. Guo, F. Hamel, H. Ninomiya and J-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. I. H. Poincaré, AN 24 (2007), 369-393.  Google Scholar

[8]

M. del Pino, M. Kowalczyk and J. Wei, On de Giorgi conjecture in dimension $N\geq9$,, Annals of Math. (to appear)., ().   Google Scholar

[9]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.  Google Scholar

[10]

R. A. Fisher, The advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Springer-Verlag, Berlin, 1983.  Google Scholar

[12]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Scient. Ec. Norm. Sup. 4ème série, t.37 (2004), 469-506.  Google Scholar

[13]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.  Google Scholar

[14]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.  Google Scholar

[15]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rat. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.  Google Scholar

[16]

F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101-123. doi: 10.3934/dcdss.2011.4.101.  Google Scholar

[17]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, AN 23 (2006), 283-329.  Google Scholar

[18]

Y. I. Kanel', Certain problems on equations in the theory of burning, Soviet. Math. Dokl., 2 (1961), 48-51.  Google Scholar

[19]

Y. I. Kanel', Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.), 59 (1962), 245-288.  Google Scholar

[20]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[21]

K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems, Phys. A, 116 (1982), 573-593. doi: 10.1016/0378-4371(82)90178-9.  Google Scholar

[22]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.  Google Scholar

[23]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II, Comm. Par. Diff. Eq., 17 (1992), 1901-1924.  Google Scholar

[24]

H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Par. Diff. Eq. , 34 (2009), 976-1002.  Google Scholar

[25]

J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines, IEEE Trans. Circuit Theory, CT-12 (1965), 400-412. Google Scholar

[26]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: Theory and applications I, GAKUTO Internat. Ser. Math. Sci. Appl., 13 (2000), 206-221.  Google Scholar

[27]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.  Google Scholar

[28]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.  Google Scholar

[29]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbfR^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651.  Google Scholar

[30]

V. Pérez-Muñuzuri, M. Gómez-Gesteira, A. P. Muñuzuri, V. A. Davydov and V. Pérez-Villar, V-shaped stable nonspiral patterns, Physical Review E, 51 (1995), 845-847. Google Scholar

[31]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, Berlin, 1984.  Google Scholar

[32]

J-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Annali di Matematica, 188 (2009), 207-233.  Google Scholar

[33]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.  Google Scholar

[34]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theoret. Population Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[35]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  Google Scholar

[36]

F. A. Smith and S. F. Pickering, Bunsen flames of unusual structure, Proceedings of the Symposium on Combustion, Vol. 1-2 (1948), 24-26. doi: 10.1016/S1062-2888(65)80006-5.  Google Scholar

[37]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.  Google Scholar

[38]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[39]

M. Taniguchi, Traveling fronts in perturbed multistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. - Supplement 2011 (The proceedings for the 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications), (to appear). Google Scholar

[40]

J. J. Tyson and P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237. doi: 10.1063/1.440418.  Google Scholar

[41]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Comm. Par. Diff. Eq., 17 (1992), 1889-1899.  Google Scholar

[42]

H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation, J. Dynam. Differential Equations, 13 (2001), 323-353. doi: 10.1023/A:1016632124792.  Google Scholar

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