# American Institute of Mathematical Sciences

January  2021, 41(1): 395-412. doi: 10.3934/dcds.2020364

## Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space

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

Received  January 2020 Revised  September 2020 Published  January 2021 Early access  October 2020

Fund Project: The author was partially supported by JSPS KAKENHI Grant Numbers JP16KT0022 and JP20H01816

The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the Allen–Cahn–Nagumo equation is studied in multi-dimensional space. Here an entire solution is meant by the solution defined for all time including negative time, even though it satisfies a parabolic partial differential equation. Especially, this equation admits traveling wave solutions connecting two stable states. It is known that there is an entire solution which behaves as two traveling wave solutions coming from both sides in one dimensional space and annihilating in a finite time and that this one-dimensional entire solution is unique up to the shift. Namely, this entire solution is symmetric with respect to some point. There is a natural question whether entire solutions coming from all directions in the multi-dimensional space are radially symmetric or not. To answer this question, radially asymmetric entire solutions will be constructed by using super-sub solutions.

Citation: Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
##### References:
 [1] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, Journal of Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E. [2] X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, Journal of Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028. [3] 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. [4] Y.-Y. Chen, H. Ninomiya and R. Taguchi, Traveling spots on multi-dimensional excitable media, J. Elliptic Parabol. Equ., 1 (2015), 281-305.  doi: 10.1007/BF03377382. [5] P. Daskalopoulos, R. Hamilton and N. Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom., 84 (2010), 455-464.  doi: 10.4310/jdg/1279114297. [6] 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. [7] 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. [8] K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154. [9] 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. [10] 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. [11] 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. [12] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ${\Bbb R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [13] J. I. Kanel, Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51. [14] H. Matano and P. Poláčik, An entire solution of a bistable parabolic equation on $\Bbb R$ with two colliding pulses, J. Funct. Anal., 272 (2017), 1956-1979.  doi: 10.1016/j.jfa.2016.11.006. [15] 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. [16] P. de Mottoni and M. Schatzman, Development of interfaces in RN, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207-220.  doi: 10.1017/S0308210500031486. [17] H. Ninomiya, Entire solutions and traveling wave solutions of the Allen–Cahn–Nagumo equation, Discrete Contin. Dyn. Syst., 39 (2019), 2001-2019. doi: 10.3934/dcds.2019084. [18] 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. [19] 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. [20] P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\Bbb R^N$: Ⅱ. Entire solutions, Comm. Partial Differential Equations, 31 (2006), 1615-1638.  doi: 10.1080/03605300600635020. [21] 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. [22] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.  doi: 10.1137/060661788. [23] 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. [24] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046.  doi: 10.3934/dcds.2012.32.1011. [25] 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. [26] 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. [27] 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. [28] 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.

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##### References:
 [1] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, Journal of Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E. [2] X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, Journal of Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028. [3] 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. [4] Y.-Y. Chen, H. Ninomiya and R. Taguchi, Traveling spots on multi-dimensional excitable media, J. Elliptic Parabol. Equ., 1 (2015), 281-305.  doi: 10.1007/BF03377382. [5] P. Daskalopoulos, R. Hamilton and N. Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom., 84 (2010), 455-464.  doi: 10.4310/jdg/1279114297. [6] 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. [7] 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. [8] K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154. [9] 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. [10] 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. [11] 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. [12] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ${\Bbb R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [13] J. I. Kanel, Some problems involving burning-theory equations, Soviet Math. Dokl., 2 (1961), 48-51. [14] H. Matano and P. Poláčik, An entire solution of a bistable parabolic equation on $\Bbb R$ with two colliding pulses, J. Funct. Anal., 272 (2017), 1956-1979.  doi: 10.1016/j.jfa.2016.11.006. [15] 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. [16] P. de Mottoni and M. Schatzman, Development of interfaces in RN, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207-220.  doi: 10.1017/S0308210500031486. [17] H. Ninomiya, Entire solutions and traveling wave solutions of the Allen–Cahn–Nagumo equation, Discrete Contin. Dyn. Syst., 39 (2019), 2001-2019. doi: 10.3934/dcds.2019084. [18] 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. [19] 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. [20] P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\Bbb R^N$: Ⅱ. Entire solutions, Comm. Partial Differential Equations, 31 (2006), 1615-1638.  doi: 10.1080/03605300600635020. [21] 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. [22] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.  doi: 10.1137/060661788. [23] 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. [24] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046.  doi: 10.3934/dcds.2012.32.1011. [25] 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. [26] 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. [27] 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. [28] 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.
The case when $p_+(t)$ is large. The dashing curve in the gray region indicates the set of $H( \boldsymbol{x})=p_+(t)$. The gray region indicates (iii). The region surrounded by the solid curve is $K_0$. One can observe that the curvature is getting small as $p_\pm$ goes to infinity
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