    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  and , 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 & Continuous Dynamical Systems - A, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205
##### References:
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show all references

##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.  Google Scholar  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.  Google Scholar  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.  Google Scholar
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