SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY

Yoshihisa Morita

doi:10.11948/2012004

2012 Volume 2 Issue 1

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Yoshihisa Morita. SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY[J]. Journal of Applied Analysis & Computation, 2012, 2(1): 57-71. doi: 10.11948/2012004

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Yoshihisa Morita. SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY[J]. Journal of Applied Analysis & Computation, 2012, 2(1): 57-71. doi: 10.11948/2012004

SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY

Yoshihisa Morita

Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan

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Abstract

Abstract

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.
- Reaction-diffusion system /
- conservation of a mass /
- equilibrium solution /
- linearized eigenvalue problem /
- Rayleigh-quotient
MSC: 35B35;35K57;49R50

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References

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