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A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition
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Critically finite random maps of an interval
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 |
We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [
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. |






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