# American Institute of Mathematical Sciences

August  2020, 40(8): 4907-4925. doi: 10.3934/dcds.2020205

## Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation

 1 Graduate School of Engineering, Musashino University, Tokyo, 135-8181, Japan 2 Department of Applied Mathematics, Waseda University, Tokyo, 169-8555, Japan 3 Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192, Japan 4 Joint Research Center for Science and Technology, Ryukoku University, Seta, Otsu, 520-2194, Japan

* Corresponding author: Shoji Yotsutani

Received  August 2019 Revised  February 2020 Published  May 2020

Fund Project: K. Kuto was supported by Grant-in-Aid. for Scientific Research (C) 19K03581. T. Tsujikawa was supported by Grant-in-Aid. for Scientific Research (C) 17K05334. S. Yotsutani was supported by Grant-in-Aid. for Scientific Research (C) 19K03593. This work was supported by Joint Research Center for Science and Technology of Ryukoku University in 2020

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [4] and [5], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

Citation: Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205
##### References:
 [1] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.  doi: 10.1080/00036817408839081. [2] X. F. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equations with a nonlocal term, Nonlinear Anal. TMA, 28 (1997), 1283-1298.  doi: 10.1016/S0362-546X(97)82875-1. [3] S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629.  doi: 10.3934/dcds.2007.19.609. [4] K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Secondary bifurcation for a nonlocal Allen-Cahn equation, J. Differential Equations, 263 (2017), 2687-2714.  doi: 10.1016/j.jde.2017.04.010. [5] K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Global solution branches for a nonlocal Allen-Cahn equation, J. Differential Equations, 264 (2018), 5928-5949.  doi: 10.1016/j.jde.2018.01.025. [6] K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 467–476. doi: 10.3934/proc.2013.2013.467. [7] Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [8] Y. Mori, A. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization, SIAM J. Appl. Math., 71 (2011), 1401-1427.  doi: 10.1137/10079118X. [9] T. Mori, K. Kuto, M. Nagayama, T. Tsujikawa and S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., (2015), 861–877. doi: 10.3934/proc.2015.0861. [10] M. Murai, W. Mastumoto and S. Yotsutani, Representation formula for the plane elastic closed curve, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 565–585. doi: 10.3934/proc.2013.2013.565. [11] M. Murai, K. Sakamoto and S. Yostutani, Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., (2015), 878–900. doi: 10.3934/proc.2015.0878. [12] R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346. [13] S. Tasaki and T. Suzuki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., TMA, 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007. [14] T. Wakasa and S. Yotsutani, Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation Ⅱ: Asymptotic formulas of eigenfunctions, J. Differential Equations, 261 (2016), 5465-5498.  doi: 10.1016/j.jde.2016.08.016.

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##### References:
 [1] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.  doi: 10.1080/00036817408839081. [2] X. F. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equations with a nonlocal term, Nonlinear Anal. TMA, 28 (1997), 1283-1298.  doi: 10.1016/S0362-546X(97)82875-1. [3] S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629.  doi: 10.3934/dcds.2007.19.609. [4] K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Secondary bifurcation for a nonlocal Allen-Cahn equation, J. Differential Equations, 263 (2017), 2687-2714.  doi: 10.1016/j.jde.2017.04.010. [5] K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Global solution branches for a nonlocal Allen-Cahn equation, J. Differential Equations, 264 (2018), 5928-5949.  doi: 10.1016/j.jde.2018.01.025. [6] K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 467–476. doi: 10.3934/proc.2013.2013.467. [7] Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [8] Y. Mori, A. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization, SIAM J. Appl. Math., 71 (2011), 1401-1427.  doi: 10.1137/10079118X. [9] T. Mori, K. Kuto, M. Nagayama, T. Tsujikawa and S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., (2015), 861–877. doi: 10.3934/proc.2015.0861. [10] M. Murai, W. Mastumoto and S. Yotsutani, Representation formula for the plane elastic closed curve, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 565–585. doi: 10.3934/proc.2013.2013.565. [11] M. Murai, K. Sakamoto and S. Yostutani, Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., (2015), 878–900. doi: 10.3934/proc.2015.0878. [12] R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346. [13] S. Tasaki and T. Suzuki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., TMA, 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007. [14] T. Wakasa and S. Yotsutani, Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation Ⅱ: Asymptotic formulas of eigenfunctions, J. Differential Equations, 261 (2016), 5465-5498.  doi: 10.1016/j.jde.2016.08.016.
Bifurcation diagram $\mathcal{S}(\mu)$
complete elliptic integrals $K(k)$, $E(k)$ and $\Pi(3/4,k)$
Graph of $\mathscr{A}(p, h)$
Bifurcation sheet $\varXi(\lambda, d)$
Bifurcation sheet on $P\!H$-plane
Bifurcation curve
Bifurcation curve on $P\!H$-plane
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