Tatsuki Mori

<p lang="fr">&lt;p style='text-indent:20px;'&gt;We are concerned with bifurcation diagrams of stationary solutions to a phase field model proposed by Fix and followed by Caginalp. We show all the global bifurcation diagrams of stationary solutions to the model in the 1-dimension case. We see that bifurcation diagrams are surprisingly rich in variety depending on the latent heat and the initial total enthalpy. For instance, bifurcation diagrams include the secondary bifurcation point where symmetric breaking occurs, and curves which connect a limit of boundary layer solutions to the other limit of internal layer solutions.&lt;/p&gt;</p>

© 2020 Elsevier Inc. We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2,3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breaking effect. This paper derives the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. To do so, we use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. And we show the instability of asymmetric solutions by the SLEP method. Finally, our results with respect to stability are supported by some numerical simulations.

The SKT cross-diffusion equation is proposed by N. Shigesada, K. Kawasaki and E. Teramoto in 1979 to investigate segregation phenomena of two competing species with each other in the same habitat area. The effect of cross-diffusion affects the population pressure between two different species. Lou and Ni derived limiting systems to see whether this effect may give rise to a spatial segregation or not, and to clarify its mechanism. In this paper, we introduce some new representation of solutions to a stationary limiting problem modified from representation by Lou, Ni and Yotsutani. We apply it to the numerical investigation of existence, non-existence, multiplicity and stability.

This paper studies the Neumann problem of a nonlocal Allen-Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure. (C) 2017 Elsevier Inc. All rights reserved.

We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.