オガワ　トシユキ   OGAWA Toshiyuki   小川　知之    所属   明治大学  総合数理学部    職種   専任教授
言語種別	英語
発行・発表の年月	2010
形態種別	学術雑誌
標題	Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass
執筆形態	共著(筆頭者以外)
掲載誌名	Nonlinearity
巻・号・頁	23(6),pp.1387-1411
著者・共著者	Yoshihisa Morita
概要	We deal with a two-component system of reaction–diffusion equations with conservation of mass in a bounded domain with the Neumann or periodic boundary conditions. This system is proposed as a conceptual model for cell polarity. Since the system has conservation of mass, the steady state problem is reduced to that of a scalar reaction–diffusion equation with a nonlocal term. That is, there is a one-to-one correspondence between an equilibrium solution of the system with a fixed mass and a solution of the scalar equation. In particular, we consider the case when the reaction term is linear in one variable. Then the equations are transformed into the same equations as the phase-field model for solidification. We thereby show that the equations allow a Lyapunov function. Moreover, by investigating the linearized stability of a nonconstant equilibrium solution, we prove that given a nondegenerate stable equilibrium solution of the nonlocal scalar equation, the corresponding equilibrium solution of the system is stable. We also exhibit global bifurcation diagrams for equilibrium solutions to specific model equations by numerics together with a normal form near a bifurcation point.