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April  2019, 39(4): 2001-2019. doi: 10.3934/dcds.2019084

Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation

Hirokazu Ninomiya ,

School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

* Corresponding author: H. Ninomiya

Received  February 2018 Published  January 2019

Fund Project: The author was partially supported by JSPS KAKENHI Grant Numbers JP26287024, JP15K04963, JP16K13778, and JP16KT0022.

In this paper, the propagation phenomena in the Allen-Cahn-Nagumo equation are considered. Especially, the relation between traveling wave solutions and entire solutions is discussed. Indeed, several types of one-dimensional entire solutions are constructed by composing one-dimensional traveling wave solutions. Combining planar traveling wave solutions provides several types of multi-dimensional traveling wave solutions. The relation between multi-dimensional traveling wave solutions and entire solutions suggests the existence of new traveling wave solutions and new entire solutions.

Keywords: Traveling wave, reaction-diffusion equations.
Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 35B40, 35C08.
Citation: Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084
References:
[1]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[2]

X. F. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

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X. F. Chen, J. S. Guo, F. Hamel, H. Ninomiya and J. M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Annales de l'Institut Henri Poincaré C: Non Linear Analysis, 24 (2007), 369-393.  doi: 10.1016/j.anihpc.2006.03.012.  Google Scholar

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Y. Y. Chen, J. S. Guo, H. Ninomiya and C. H. Yao, Entire solutions originating from monotone fronts to the Allen-Cahn equation, Physica D, 378/379 (2018), 1-19. doi: 10.1016/j.physd.2018.04.003.  Google Scholar

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M. del Pino, M. Kowalczyk and J. Wei, Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.  doi: 10.1002/cpa.21438.  Google Scholar

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

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Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454.  Google Scholar

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J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. System, 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

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K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

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

[11]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Systems, 14 (2006), 75-92.  doi: 10.3934/dcds.2006.14.75.  Google Scholar

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F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

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

[14]

Y. I. Kanel, Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51.   Google Scholar

[15]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sec. A, 1 (1937), 1-26.   Google Scholar

[16]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[17]

Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sin.(NS), 3 (2008), 567-584.   Google Scholar

[18]

H. Ninomiya, Multi-dimensional entire solutions of the Allen-Cahn-Nagumo equation, in preparation. Google Scholar

[19]

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

[20]

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

[21]

P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $ \mathbb{R} ^N$: Ⅱ. entire solutions, Communications in Partial Differential Equations, 31 (2006), 1615-1638.  doi: 10.1080/03605300600635020.  Google Scholar

[22]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[23]

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

[24]

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

[25]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion, Discrete Contin. Dyn. System, 32 (2012), 1011-1046.  doi: 10.3934/dcds.2012.32.1011.  Google Scholar

[26]

M. Taniguchi, An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen–Cahn equation, SIAM J. Math. Anal., 47 (2015), 455-476.  doi: 10.1137/130945041.  Google Scholar

[27]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[28]

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

[29]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

show all references

References:
[1]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[2]

X. F. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[3]

X. F. Chen, J. S. Guo, F. Hamel, H. Ninomiya and J. M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Annales de l'Institut Henri Poincaré C: Non Linear Analysis, 24 (2007), 369-393.  doi: 10.1016/j.anihpc.2006.03.012.  Google Scholar

[4]

Y. Y. Chen, J. S. Guo, H. Ninomiya and C. H. Yao, Entire solutions originating from monotone fronts to the Allen-Cahn equation, Physica D, 378/379 (2018), 1-19. doi: 10.1016/j.physd.2018.04.003.  Google Scholar

[5]

M. del Pino, M. Kowalczyk and J. Wei, Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.  doi: 10.1002/cpa.21438.  Google Scholar

[6]

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

[7]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454.  Google Scholar

[8]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. System, 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[9]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

[10]

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

[11]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Systems, 14 (2006), 75-92.  doi: 10.3934/dcds.2006.14.75.  Google Scholar

[12]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[13]

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

[14]

Y. I. Kanel, Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51.   Google Scholar

[15]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sec. A, 1 (1937), 1-26.   Google Scholar

[16]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[17]

Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sin.(NS), 3 (2008), 567-584.   Google Scholar

[18]

H. Ninomiya, Multi-dimensional entire solutions of the Allen-Cahn-Nagumo equation, in preparation. Google Scholar

[19]

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

[20]

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

[21]

P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $ \mathbb{R} ^N$: Ⅱ. entire solutions, Communications in Partial Differential Equations, 31 (2006), 1615-1638.  doi: 10.1080/03605300600635020.  Google Scholar

[22]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[23]

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

[24]

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

[25]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion, Discrete Contin. Dyn. System, 32 (2012), 1011-1046.  doi: 10.3934/dcds.2012.32.1011.  Google Scholar

[26]

M. Taniguchi, An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen–Cahn equation, SIAM J. Math. Anal., 47 (2015), 455-476.  doi: 10.1137/130945041.  Google Scholar

[27]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[28]

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

[29]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

Figure 1.  Profiles of supersolutions for one-dimensional entire solutions when $ -t $ is large, where $ f(u) = u(1-u)(u-1/12) $
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Figure 2.  Profiles of supersolutions for one-dimensional entire solutions when $ -t $ is large, where $ f(u) = u(1-u)(u-1/12) $
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Figure 3.  Contours of $ V_1^- $
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Figure 4.  Zipping wave solution when $ f(u) = u(1-u)(u-1/2) $
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Figure 5.  Example of $ \Gamma_0 $ and the equidistant curves from $ K_0 $ in $ \mathbb{R} ^2 $
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