Workshop Program:

09:40 – 09:45 Opening

09:45 – 10:25 Masayasu Mimura

10:30 – 11:10 Yoshihisa Morita

11:15 – 11:55 Chia-Yu Hsieh

11:55 – 13:00 Lunch

13:00 – 13:40 Shoji Yotsutani

13:45 – 14:25 Kota Ikeda

14:25 – 14:40 Break

14:40 – 15:20 Jann-Long Chern

15:25 – 16:05 Tomoyuki Miyaji

Jann-Long Chern(National Central University)

Title:

Uniqueness of positive solution to some coupled cooperative variational elliptic systems on an interval

Abstract:

Oscillatory behavior of solutions of linearized equations for cooperative semilinear elliptic systems of two equations on one-dimensional domains are proved, and it is shown that the stability of the positive solutions for such semilinear system is closely related to the oscillatory behavior. These properties are used to prove the uniqueness of positive solutions to some semilinear elliptic systems with nonlinearities satisfying certain variational structure and growth conditions. This is a joint work with Yulian An and Junping Shi.

Chia-Yu Hsieh(National Taiwan University)

Title:

Stability of boundary layer solutions of Poisson-Nernst-Planck systems

Abstract:

The Poisson-Nernst-Planck (PNP) system has been widely used to describe the electron transport of semiconductors and the ion transport of ionic solutions, and plays a crucial role in the study of many physical and biological problems. If the Robin boundary condition is imposed for the electrostatic potential, the PNP system admits a boundary layer solution as a steady state. We study the stability of boundary layer solutions of PNP system. By transforming the perturbed problem into another parabolic system with a new and useful energy law, we prove that the $H^{-1}$-norm of the solution of the perturbed problem decays exponentially.

Kota Ikeda(Meiji University)

Title:

Reduction approach to a reaction-diffusion system with the delta function

–Collective Motion of Camphor Boats –

Abstract:

The collective motion of camphor boats in the water channel exhibits both a homogeneous and an inhomogeneous state, depending on the number of boats [3]. The motion of each camphor boat is described by a traveling pulse in a reaction-diffusion model proposed in [2], in which camphor boats are assumed to be interact each other by the change of surface tension by diffusive molecules on the water surface. In order to verify the inhomogeneous motion of camphor boats, we have to study the linearized eigenvalue problem and see the destabilization of the homogeneous flow. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Then we would like to derive a reduced system from the original model and analyze it by applying the center manifold theorem.

Several reaction-diffusion systems can generate a solution with a pulse shape. The authors in [1] treat pulse-pulse interaction mathematically, and derive reduced systems of an ODE form from reaction-diffusion models by applying a center manifold theorem. Since the delta functions naturally arise in our model, the theory established in L2-framework cannot be applied directly. In this talk, we modify the previous results in [1] and propose a new approach of reduction to systems with the delta function.

This is a joint work with Ei Shin-Ichiro in Hokkaido University.

References

[1] S.-I. Ei, M. Mimura, M. Nagayama: Pulse-pulse interaction in reaction-diffusion systems.

Phys. D. 165 (2002), 176–198.

[2] M. Nagayama, S. Nakata, Y. Doi, Y. Hayashima: A theoretical and experimental study on the unidirectional motion of a camphor disk. Phys. D. 194 (2004), 151–165.

[3] N. J. Suematsu, S. Nakata, A. Awazu, H. Nishimori: Collective behavior of inanimate boats. Physical Review E 81 (2010), 056210.

Masayasu Mimura(MIMS/Meiji University)

Title:

2D-traveling waves in an Allen-Cahn equation with chemotaxis

Abstract:

We consider an Allen-Cahn equation with chemotactic effect. By using the singular perturbation analysis, the existence and stability of 1D traveling front and standing pulse solutions are shown. Moreover, with the help of numerical procedures, we show that the interaction of planar fronts or pulses generates truly 2D traveling waves which is called wedge- and zipper-shaped, depending on the kind interaction between two fronts or pulses.

Tomoyuki Miyaji(Meiji University)

Title:

Structure and bifurcation of a four-dimensional dynamical system arising from spot dynamics in planar reaction-diffusion systems

Abstract:

We study a four-dimensional dynamical system defined by a system of ordinary differential equations which is derived through the center manifold reduction for a reaction-diffusion system. It describes a spot dynamics near a bifurcation point where a stationary spot loses stability and a traveling spot bifurcates. In particular, we consider the traveling spot in a rectangular domain with the Neumann boundary conditions. We apply methods of dynamical systems and bifurcation theory for understanding periodic and chaotic behavior of the system.

Yoshihisa Morita(Ryukoku University)

Title:

Standing fronts of the FitzHugh-Nagumo system and their interacting dynamics

Abstract:

We deals with the FitzHugh-Nagumo activator-inhibitor system on the infinite interval. When the degradation rate of the inhibitor $v$ is large, the system allows a stable standing wavefront. It is shown that depending on the diffusion constant there are three types of the asymptotic behavior of the wavefront at the infinity. Then we consider the two facing front-dynamics and show that the dynamics can be classified as repulsive, attractive and stationary corresponding to the three types. This talk is based on the recent joint work with C.-N. Chen (Natinal Tsing Hua Univ.) and S. Ei (Hokkaido Univ.).

Shoji Yotsutani(Ryukoku University)

Title:

Secondary bifurcation for a nonlocal Allen-Cahn equation

Abstract:

We study a Neumann problem of a nonlocal Allen-Cahn equation in a finite interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curveof solutions with odd-symmetry. Our proof is based ona 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 results concerning the global bifurcation structure. This is a joint work with K. Kuto, T. Mori, and T. Tsujikawa.