OGAWA Toshiyuki    Department   Undergraduate School  , School of Interdisciplinary Mathematical Sciences    Position   Professor
Language	English
Publication Date	2010
Type	Academic Journal
Title	Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass
Contribution Type	Co-authored (other than first author)
Journal	Nonlinearity
Volume, Issue, Page	23(6),pp.1387-1411
Author and coauthor	Yoshihisa Morita
Details	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.