Asymptotic behavior of equilibrium states of reaction–diffusion systems with mass conservation
Asymptotic behavior of equilibrium states of reaction–diffusion systems with mass conservation
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DOI：
10.1016/j.jde.2017.09.015
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发表时间：
2018-01
影响因子：
2.4
通讯作者：
J. Chern;Y. Morita;T. Shieh
中科院分区：
文献类型：
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作者：
J. Chern;Y. Morita;T. Shieh
我们用诺依曼边界条件下的守恒定律来处理反应扩散系统的平稳问题。结果表明，平稳问题变成了具有质量约束的能量泛函的欧拉-拉格朗日方程。当域为有限区间 (0, 1) 时，我们研究能量的严格单调极小值的渐近轮廓，因为 d（系统扩散系数之比）趋于零。鉴于势首项中的对数函数，我们得到满足关系式 ε:= d= log κ/κ 2 的标度参数 κ。主要结果表明，最小化序列收敛到狄拉克质量乘以总质量，并且通过使用 κ 进行标度，渐近轮廓在非零区域中呈现抛物线。我们还证明了当质量很小时存在不稳定的单调解。
We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neumann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coefficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κ satisfying the relation ε:= d= log κ/κ 2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κ the asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.