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

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

Tatsuki Mori 1, , Kousuke Kuto 2, , Tohru Tsujikawa 3, and  Shoji Yotsutani 4,,

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.

Keywords: Allen-Cahn equation, nonlocal term, exact solution.
Mathematics Subject Classification: Primary: 34C23, 34B10; Secondary: 37G40.
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.

show all references

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.

Figure 1.  Bifurcation diagram $ \mathcal{S}(\mu) $
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Figure 2.  complete elliptic integrals $ K(k) $, $ E(k) $ and $ \Pi(3/4,k) $
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Figure 3.  Graph of $ \mathscr{A}(p, h) $
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Figure 4.  Bifurcation sheet $ \varXi(\lambda, d) $
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Figure 5.  Bifurcation sheet on $ P\!H $-plane
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Figure 6.  Bifurcation curve
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Figure 7.  Bifurcation curve on $ P\!H $-plane
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