doi: 10.3934/cpaa.2022116
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Large time behavior of the solutions with spreading fronts in the Allen-Cahn equations on $ \mathbb R^n $

Faculty of Science and Engineering, Iwate University, Ueda 3-18-34, Morioka, Iwate, 020-8550, Japan

Received  October 2021 Revised  June 2022 Early access July 2022

Fund Project: The author is partially supported by Grant-in-Aid for Scientific Research (C) (19K03556)

We consider the initial value problem of the Allen-Cahn equation on $ \mathbb R^n $ with $ n\geq2 $ and study the large time behavior of the solutions with spreading fronts. Our result states that, under some mild assumptions on initial values, the solution develops a well-formed front whose position roughly coincides with the spreading sphere that is a solution of mean curvature flow with a driving constant, and that in each radial direction the distance between the level set of the solution and the spreading sphere converges to a value as time goes to infinity.

Citation: Mitsunori Nara. Large time behavior of the solutions with spreading fronts in the Allen-Cahn equations on $ \mathbb R^n $. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022116
References:
[1]

M. AlfaroH. GarckeD. HilhorstH. Matano and R. Schätzle, Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen–Cahn equation, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 673-706.  doi: 10.1017/S0308210508000541.

[2]

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

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.

[4]

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

[5]

C. M. Elliott and R. Schätzle, The limit of the anisotropic double-obstacle Allen-Cahn equation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1217-1234.  doi: 10.1017/S0308210500023374.

[6]

C. M. Elliott and R. Schätzle, The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the nonsmooth case, SIAM J. Math. Anal., 28 (1997), 274-303.  doi: 10.1137/S0036141095286733.

[7]

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

[8]

Y. GigaT. Ohtsuka and R. Schätzle, On a uniform approximation of motion by anisotropic curvature by the Allen-Cahn equations, Interfaces Free Bound., 8 (2006), 317-348.  doi: 10.4171/IFB/146.

[9]

A. HoffmanH. J. Hupkes and E. S. Van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions, Trans. Amer. Math. Soc., 367 (2015), 8757-8808.  doi: 10.1090/S0002-9947-2015-06392-2.

[10]

A. Hoffman, H. J. Hupkes and E. S. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, Mem. Amer. Math. Soc., 250, 2017. doi: 10.1090/memo/1188.

[11]

M. Jukić and H. J. Hupkes, Curvature-driven front propagation through planar lattices in oblique directions, Commun. Pure Appl. Anal., 21 (2022), 2189-2251.  doi: 10.3934/cpaa.2022055.

[12]

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

[13]

K. R. T. Jones Christopher, Spherically symmetric solutions of a reaction-diffusion equation, J. Differ. Equ., 49 (1983), 142-169.  doi: 10.1016/0022-0396(83)90023-2.

[14]

K. R. T. Jones Christopher, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364.  doi: 10.1216/RMJ-1983-13-2-355.

[15]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II, Commun. Partial Differ. Equ., 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908.

[16]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, R.I., 1968.

[17]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[18]

H. MatanoY. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equations on $ \mathbb R^n$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003.

[19]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differ. Equ., 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029.

[20]

H. MatanoM. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equations, Commun. Partial Differ. Equ., 34 (2009), 976-1002.  doi: 10.1080/03605300902963500.

[21]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8.

[22]

M. Taniguchi, Traveling Front Solutions in Reaction-Diffusion Equations, MSJ Memoirs, 39 Mathematical Society of Japan, Tokyo, 2021. doi: 10.1142/e070.

[23]

K. Uchiyama, Asymptotic behavior of solutions of reaction–diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311.  doi: 10.1007/BF00276293.

[24]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Commun. Partial Differ. Equ., 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.

[25]

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

show all references

References:
[1]

M. AlfaroH. GarckeD. HilhorstH. Matano and R. Schätzle, Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen–Cahn equation, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 673-706.  doi: 10.1017/S0308210508000541.

[2]

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

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.

[4]

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

[5]

C. M. Elliott and R. Schätzle, The limit of the anisotropic double-obstacle Allen-Cahn equation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1217-1234.  doi: 10.1017/S0308210500023374.

[6]

C. M. Elliott and R. Schätzle, The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the nonsmooth case, SIAM J. Math. Anal., 28 (1997), 274-303.  doi: 10.1137/S0036141095286733.

[7]

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

[8]

Y. GigaT. Ohtsuka and R. Schätzle, On a uniform approximation of motion by anisotropic curvature by the Allen-Cahn equations, Interfaces Free Bound., 8 (2006), 317-348.  doi: 10.4171/IFB/146.

[9]

A. HoffmanH. J. Hupkes and E. S. Van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions, Trans. Amer. Math. Soc., 367 (2015), 8757-8808.  doi: 10.1090/S0002-9947-2015-06392-2.

[10]

A. Hoffman, H. J. Hupkes and E. S. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, Mem. Amer. Math. Soc., 250, 2017. doi: 10.1090/memo/1188.

[11]

M. Jukić and H. J. Hupkes, Curvature-driven front propagation through planar lattices in oblique directions, Commun. Pure Appl. Anal., 21 (2022), 2189-2251.  doi: 10.3934/cpaa.2022055.

[12]

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

[13]

K. R. T. Jones Christopher, Spherically symmetric solutions of a reaction-diffusion equation, J. Differ. Equ., 49 (1983), 142-169.  doi: 10.1016/0022-0396(83)90023-2.

[14]

K. R. T. Jones Christopher, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364.  doi: 10.1216/RMJ-1983-13-2-355.

[15]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II, Commun. Partial Differ. Equ., 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908.

[16]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, R.I., 1968.

[17]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[18]

H. MatanoY. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equations on $ \mathbb R^n$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003.

[19]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differ. Equ., 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029.

[20]

H. MatanoM. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equations, Commun. Partial Differ. Equ., 34 (2009), 976-1002.  doi: 10.1080/03605300902963500.

[21]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8.

[22]

M. Taniguchi, Traveling Front Solutions in Reaction-Diffusion Equations, MSJ Memoirs, 39 Mathematical Society of Japan, Tokyo, 2021. doi: 10.1142/e070.

[23]

K. Uchiyama, Asymptotic behavior of solutions of reaction–diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311.  doi: 10.1007/BF00276293.

[24]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Commun. Partial Differ. Equ., 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.

[25]

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

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