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October  2022, 42(10): 4707-4740. doi: 10.3934/dcds.2022069

## Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $\mathit{\boldsymbol{\mathbb{R}^N}}$

 1 Department of Mathematics and Physics, Faculty of Science, Kanagawa University, 2946 Tsuchiya, Hiratsuka City, Kanagawa, 259-1293, Japan 2 Faculty of Science and Engineering, Iwate University, Ueda 3-18-34, Morioka, Iwate, 020-8550, Japan

*Corresponding author: Hiroshi Matsuzawa

Received  September 2021 Revised  April 2022 Published  October 2022 Early access  May 2022

Fund Project: H. Matsuzawa was partially supported by Grant-in-Aid for Scientific Research (C) (20K03709), and M. Nara was partially supported by Grant-in-Aid for Scientific Research (C) (19K03556)

We consider the Cauchy problem for an anisotropic reaction diffusion equation with a multi-stable nonlinearity on $\mathbb{R}^N$, $N\ge 2$ and investigate the large time behavior of the solution. This problem with a bistable nonlinearity has been investigated by Matano, Mori and Nara [Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), pp. 585-626]. They showed that under the suitable condition on the initial function the solution develops a spreading front whose position is closely approximated by the expanding Wulff shape for all large times. In this paper we will extend their results to the problem with a multi-stable type nonlinearity, that is, the case where the nonlinearity can be decomposed to $K$ number of bistable nonlinearities and show that under certain conditions on the nonlinearity and the initial function the solution develops $K$ number of spreading fronts whose positions are closely approximated by the expanding Wulff shapes with different expanding speeds. In other words, for any direction the solution on the ray of the direction looks like stacked traveling waves, that is, on the ray the solution approaches the so called propagating terrace. The key step for extension to multi-stable case is to construct $K$ number of upper solutions and lower solutions all at once.

Citation: Hiroshi Matsuzawa, Mitsunori Nara. Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $\mathit{\boldsymbol{\mathbb{R}^N}}$. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4707-4740. doi: 10.3934/dcds.2022069
##### References:
 [1] M. Alfaro, H. Garcke, D. Hilhorst, H. 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, Bounds for the fundamental solution of a parabolic equation, Bull. Am. Math. Soc., 73 (1967), 890-896.  doi: 10.1090/S0002-9904-1967-11830-5. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, Lecture Notes in Math. Springer, Berlin, 446 (1975), 5–49. [4] 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. [5] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In Perspectives in Nonlinear Partial Differential Equations. In Honor of H. Brezis, Contemp. Math., 446 (2007), 101–123. doi: 10.1090/conm/446/08627. [6] W. Ding and H. Matano, Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$, SIAM J. Math. Anal., 52 (2020), 2411-2462.  doi: 10.1137/19M1268987. [7] Y. Du and H. Matano, Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over $\mathbb{R}^N$, preprint, 2017, arXiv: 1711.00952. [8] A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), 5541-5566.  doi: 10.1090/S0002-9947-2014-06105-9. [9] 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. [10] 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. [11] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [12] Y. Giga, T. 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. [13] T. Gilleti and L. Rossi, Pulsating solutions for multidimensional bistable and multistable equations, Math. Ann., 378 (2020), 1555-1611.  doi: 10.1007/s00208-019-01919-z. [14] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970. [15] Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II : Asymptotic profiles of solutions and radial terrace solution, preprint. [16] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. [17] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302. [18] H. Matano, Y. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equation on $\mathbb{R}^N$, Ann. I. H. Poincaré Anal. Non Linéaire, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003. [19] P. Poláčik, Planar propagating terrace and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations, SIAM J. Math. Anal., 49 (2017), 3716-3740.  doi: 10.1137/16M1100745. [20] P. Poláčik, Propagating terrace and the dynamics of front-like solutions of reaction-diffusion equations on $\mathbb{R}$, Mem. Amer. Math. Soc., 264 (2020). [21] E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. I. H. Poincaré Anal. Non Linéaire, 25 (2008), 381-424.  doi: 10.1016/j.anihpc.2006.12.005.

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##### References:
 [1] M. Alfaro, H. Garcke, D. Hilhorst, H. 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, Bounds for the fundamental solution of a parabolic equation, Bull. Am. Math. Soc., 73 (1967), 890-896.  doi: 10.1090/S0002-9904-1967-11830-5. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, Lecture Notes in Math. Springer, Berlin, 446 (1975), 5–49. [4] 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. [5] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In Perspectives in Nonlinear Partial Differential Equations. In Honor of H. Brezis, Contemp. Math., 446 (2007), 101–123. doi: 10.1090/conm/446/08627. [6] W. Ding and H. Matano, Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$, SIAM J. Math. Anal., 52 (2020), 2411-2462.  doi: 10.1137/19M1268987. [7] Y. Du and H. Matano, Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over $\mathbb{R}^N$, preprint, 2017, arXiv: 1711.00952. [8] A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), 5541-5566.  doi: 10.1090/S0002-9947-2014-06105-9. [9] 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. [10] 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. [11] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [12] Y. Giga, T. 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. [13] T. Gilleti and L. Rossi, Pulsating solutions for multidimensional bistable and multistable equations, Math. Ann., 378 (2020), 1555-1611.  doi: 10.1007/s00208-019-01919-z. [14] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970. [15] Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II : Asymptotic profiles of solutions and radial terrace solution, preprint. [16] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. [17] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302. [18] H. Matano, Y. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equation on $\mathbb{R}^N$, Ann. I. H. Poincaré Anal. Non Linéaire, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003. [19] P. Poláčik, Planar propagating terrace and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations, SIAM J. Math. Anal., 49 (2017), 3716-3740.  doi: 10.1137/16M1100745. [20] P. Poláčik, Propagating terrace and the dynamics of front-like solutions of reaction-diffusion equations on $\mathbb{R}$, Mem. Amer. Math. Soc., 264 (2020). [21] E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. I. H. Poincaré Anal. Non Linéaire, 25 (2008), 381-424.  doi: 10.1016/j.anihpc.2006.12.005.
(Left) top view of a spreading front with $K = 2$, and the cross section of the solution that approaches at large time to the profile of a propagating terrace (stacked traveling waves) along the direction ${\boldsymbol{\nu}}$. (Right) A planar propagating terrace whose level set is tangential to the spreading Wulff shapes, and its cross section
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