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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April
2019, 39(4): 2001-2019.
doi: 10.3934/dcds.2019084
Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation
School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan |
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
K. P. Hadeler and F. Rothe,
Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
Y. I. Kanel,
Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51.
|
| [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.
|
| [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. |
| [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.
|
| [18] |
H. Ninomiya, Multi-dimensional entire solutions of the Allen-Cahn-Nagumo equation, in preparation. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
K. P. Hadeler and F. Rothe,
Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
Y. I. Kanel,
Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51.
|
| [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.
|
| [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. |
| [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.
|
| [18] |
H. Ninomiya, Multi-dimensional entire solutions of the Allen-Cahn-Nagumo equation, in preparation. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
Profiles of supersolutions for one-dimensional entire solutions when $ -t $ is large, where $ f(u) = u(1-u)(u-1/12) $
Figure 3.
Contours of $ V_1^- $
Figure 4.
Zipping wave solution when $ f(u) = u(1-u)(u-1/2) $
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