Representation formulas of solutions and bifurcation sheets to a nonlocal allen-cahn equation

Tatsuki Mori; Kousuke Kuto; Tohru Tsujikawa; Shoji Yotsutani

doi:10.3934/dcds.2020205

Representation formulas of solutions and bifurcation sheets to a nonlocal allen-cahn equation

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	4907-4925
Number of pages	19
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	40
Issue number	8
DOIs	https://doi.org/10.3934/dcds.2020205
Publication status	Published - 2020 Aug

Keywords

Allen-Cahn equation
Exact solution
Nonlocal term

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2020205

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title = "Representation formulas of solutions and bifurcation sheets to a nonlocal allen-cahn equation",

abstract = "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, Exact solution, Nonlocal term",

author = "Tatsuki Mori and Kousuke Kuto and Tohru Tsujikawa and Shoji Yotsutani",

note = "Funding Information: 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. ∗ Corresponding author: Shoji Yotsutani. Publisher Copyright: {\textcopyright} 2020 American Institute of Mathematical Sciences. All rights reserved.",

year = "2020",

month = aug,

doi = "10.3934/dcds.2020205",

language = "English",

volume = "40",

pages = "4907--4925",

journal = "Discrete and Continuous Dynamical Systems- Series A",

issn = "1078-0947",

publisher = "Southwest Missouri State University",

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T1 - Representation formulas of solutions and bifurcation sheets to a nonlocal allen-cahn equation

AU - Mori, Tatsuki

AU - Kuto, Kousuke

AU - Tsujikawa, Tohru

AU - Yotsutani, Shoji

N1 - Funding Information: 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. ∗ Corresponding author: Shoji Yotsutani. Publisher Copyright: © 2020 American Institute of Mathematical Sciences. All rights reserved.

PY - 2020/8

Y1 - 2020/8

N2 - 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.

AB - 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.

KW - Allen-Cahn equation

KW - Exact solution

KW - Nonlocal term

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VL - 40

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JF - Discrete and Continuous Dynamical Systems- Series A

IS - 8

ER -

Representation formulas of solutions and bifurcation sheets to a nonlocal allen-cahn equation

Abstract

Keywords

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