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

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
  Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127  Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012  Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703  Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473  Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823  Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015  Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025  Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679  Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577  Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319  Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009  Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024  Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407  Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077  Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517  Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113  Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391  Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111  Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159  Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

2020 Impact Factor: 1.392