Singular limit system and pattern formation of some reactiondiffusion system
Project/Area Number  10640143 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  MIYAZAKI UNIVERSITY 
Principal Investigator 
TSUJIKAWA Tohru Miyazaki University, Department of Engineering, Professor, 工学部, 教授 (10258288)

Project Period (FY) 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥3,700,000 (Direct Cost : ¥3,700,000)
Fiscal Year 2001 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  Reaction diffusion equation / Singular perturbation / Exponential attractor / Squeezing property / REACTIONDIFFUSION EQUATION / SIHGULAR LIMIT SYSTEM / GLOBAL ATTRACTOR / INTER FACE EQUATION / 反応拡散方程式 / 非平衡力学系 / 特異極限 / 進行波解 / 縮約系 / 特異極限解析 / ケラー・ジーゲル モデル 
Research Abstract 
(1) A chemotaxisgrowth model and an absorbateinduced phase transition model are able to treat in the framework of the reactiondiffusion system with advection terms. The existence of the local solution of these systems is proved in the general semigroup theory. By using the a priori estimate and the comparison theorem, it is shown that the global solutions of these system exist in the suitable functional space. Moreover, we prove that the dimension of the exponential attractor, which is some kind of property governed the dynamics of the system due to Temam et. al, is finite by showing the squeezing property. (2) It is generally difficult to determine the dimension of the exponential attractor except for the reaction diffusion equation which is not a system, and the one space dimension of the considered domain. For the first step to show it, we consider the existence of stationary solutions and traveling solutions and their stability. We prove these of the planar pulse stationary solutions, planar traveling front solutions and radial symmetric pulse stationary solutions of the chemotaxisgrowth model in 2dimensional plane. Showing the stability of these solutions, we must estimate the distribution of the eigenvalues of the linearized eigenvalue problem of the system. This problem is solved by the singular limit analysis because that the diffusion coefficient of the system is small. We first show that for the small coefficient t he dominant term of the eigenvalues determined the stability is corresponding to the coefficient of the linear differential ordinary equation due to the singular limit system, which is obtained by the reduction of the original reaction diffusion system. From these results, we will have the singular limit system of the absorbateinduced phase transition model because of the smallness of the diffusion coefficient.

Report
(5results)
Research Output
(11results)