Global solution branches for a nonlocal Allen–Cahn equation

Kousuke Kuto; Tatsuki Mori; Tohru Tsujikawa; Shoji Yotsutani

doi:10.1016/j.jde.2018.01.025

Global solution branches for a nonlocal Allen–Cahn equation

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

^*Corresponding author for this work

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

5 Citations (Scopus)

Abstract

We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.

Original language	English
Pages (from-to)	5928-5949
Number of pages	22
Journal	Journal of Differential Equations
Volume	264
Issue number	9
DOIs	https://doi.org/10.1016/j.jde.2018.01.025
Publication status	Published - 2018 May 5
Externally published	Yes

Keywords

Allen–Cahn equation
Bifurcation
Level set analysis
Monotonicity
Nonlocal term

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2018.01.025

Cite this

@article{76ba936d850b4bf387c12e23d24e42e5,

title = "Global solution branches for a nonlocal Allen–Cahn equation",

abstract = "We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.",

keywords = "Allen–Cahn equation, Bifurcation, Level set analysis, Monotonicity, Nonlocal term",

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

note = "Funding Information: K. Kuto was supported by Grant-in-Aid for Scientific Research (C) 15K04948. 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) 15K04972. This work was supported by the Joint Research Center for Science and Technology of Ryukoku University in 2017. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = may,

day = "5",

doi = "10.1016/j.jde.2018.01.025",

language = "English",

volume = "264",

pages = "5928--5949",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

number = "9",

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TY - JOUR

T1 - Global solution branches for a nonlocal Allen–Cahn equation

AU - Kuto, Kousuke

AU - Mori, Tatsuki

AU - Tsujikawa, Tohru

AU - Yotsutani, Shoji

N1 - Funding Information: K. Kuto was supported by Grant-in-Aid for Scientific Research (C) 15K04948. 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) 15K04972. This work was supported by the Joint Research Center for Science and Technology of Ryukoku University in 2017. Publisher Copyright: © 2018 Elsevier Inc.

PY - 2018/5/5

Y1 - 2018/5/5

N2 - We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.

AB - We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.

KW - Allen–Cahn equation

KW - Bifurcation

KW - Level set analysis

KW - Monotonicity

KW - Nonlocal term

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U2 - 10.1016/j.jde.2018.01.025

DO - 10.1016/j.jde.2018.01.025

M3 - Article

AN - SCOPUS:85040638952

SN - 0022-0396

VL - 264

SP - 5928

EP - 5949

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 9

ER -

Global solution branches for a nonlocal Allen–Cahn equation

Abstract

Keywords

ASJC Scopus subject areas

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