SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY
SPECTRUM COMPARISON FOR A CONSERVED REACTION-DIFFUSION SYSTEM WITH A VARIATIONAL PROPERTY
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DOI：
10.11948/2012004
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发表时间：
2012-02
影响因子：
1.1
通讯作者：
Y. Morita
中科院分区：
文献类型：
--
作者：
Y. Morita
我们处理的是一个两分量反应扩散方程组，它的质量守恒在一个有界区域内，受Neumann或周期边界条件的约束。我们考虑守恒系统转化为相场型系统的情况。然后将定常问题归结为一个带有非局部项的标量反应扩散方程。我们研究了系统平衡解的线性化特征值问题，并用Rayleigh商将其与标量非局部方程的线性化问题的特征值进行了比较。主要定理告诉我们，这些问题的负本征值的个数是一致的。因此，标量非局部反应扩散方程非常数解的稳定性结果适用于该系统。
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.