## Applied Analysis

Seminar information archive ～09/11｜Next seminar｜Future seminars 09/12～

Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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### 2007/11/08

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

弱い飽和効果をもったGierer-Meinhardt systemにおける軸対称領域上での多重ピーク解の構成と漸近挙動について

**倉田 和浩**(首都大学東京・理工学研究科・数理情報科学専攻)弱い飽和効果をもったGierer-Meinhardt systemにおける軸対称領域上での多重ピーク解の構成と漸近挙動について

[ Abstract ]

This talk is based on the joint work with Kotaro Morimoto (Tokyo Metropolitan University).

We are concerned with stationary solutions to the following reaction diffusion system which is called the Gierer-Meinhardt system with saturation.

$A_t=\\epsilon^2 \\Delta A-A+A^2/(H(1+kA^2), A>0,$

$\\tau H_t=D\\Delta H-H+A2, H>0,$

where $\\epsilon >0$, $\\tau \\geq 0$, $k>0$.

The unknowns $A$ and $H$ represent the concentrations of the activator and the inhibitor. Here $\\Omega$ is a bounded smooth domain in $R^N$ and we consider homogeneous Neumann boundary conditions. When $\\Omega$ is an $x_N$-axially symmetric domain and $2\\leq N\\leq 5$, for sufficiently small $\\epsilon>0$ and large $D>0$, we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of $x^N$-axis and $\\partial \\Omega$, under the condition that $k\\epsilon^{-2N}$ converges to some $k_0\\in[0,\\infty)$ as $\\epsilon \\to 0$.

In my talk, I will explain related results comparing the differences between the case $k=0$ and $k>0$, the basic strategy of the proof of our results with some details, and open questions.

This talk is based on the joint work with Kotaro Morimoto (Tokyo Metropolitan University).

We are concerned with stationary solutions to the following reaction diffusion system which is called the Gierer-Meinhardt system with saturation.

$A_t=\\epsilon^2 \\Delta A-A+A^2/(H(1+kA^2), A>0,$

$\\tau H_t=D\\Delta H-H+A2, H>0,$

where $\\epsilon >0$, $\\tau \\geq 0$, $k>0$.

The unknowns $A$ and $H$ represent the concentrations of the activator and the inhibitor. Here $\\Omega$ is a bounded smooth domain in $R^N$ and we consider homogeneous Neumann boundary conditions. When $\\Omega$ is an $x_N$-axially symmetric domain and $2\\leq N\\leq 5$, for sufficiently small $\\epsilon>0$ and large $D>0$, we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of $x^N$-axis and $\\partial \\Omega$, under the condition that $k\\epsilon^{-2N}$ converges to some $k_0\\in[0,\\infty)$ as $\\epsilon \\to 0$.

In my talk, I will explain related results comparing the differences between the case $k=0$ and $k>0$, the basic strategy of the proof of our results with some details, and open questions.