| Abstract: |
| We consider the Newmann problem of a 1D stationary Allen-Cahn equation with nonlocal term.
This equation is a typical model of the limiting system of reaction-diffusion systems.
For example, a diffusion coefficient of the system tends to infinity, which is called a shadow system.
It is shown that the global behavior of the branch of asymmetric solutions which bifurcate from a point on the branch
of odd-symmetric solutions. This solutions also bifurcate from a trivial solution.
It is difficult to show the appearance of the secondary bifurcation point due to be apart from the first bifurcation point.
The method using the asymptotic analysis and the complete elliptic integrals proves the existence and uniqueness of secondary bifurcation point.
The proof of the global behavior of the bifurcation branch is based on a levelset analysis for an integral map associated with the nonlocal term.
Moreover, we explain that the stability of the symmetric solutions loses at the secondary bifurcation point.
The global structure of bifurcation branches is shown by some numerical simulations.
This talk is based on joint work with Prof. K. Kuto (Univ. Electro Commu.), Prof. Y. Miyamoto (Univ. of Tokyo), Prof. T. Mori (Osaka Univ.), Prof. S. Yotsutani (Ryukoku Univ.). |
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