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Volume 2, Number 1, 2012, Pages 57-71                                                                DOI:10.11948/2012004
Spectrum comparison for a conserved reaction-diffusion system with a variational property
Yoshihisa Morita
Keywords:Reaction-diffusion system
      We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.
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