# American Institute of Mathematical Sciences

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 [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.

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

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