Asymptotic behavior of equilibrium states of reaction-diffusion systems with mass conservation

作者：Chern Jann Long; Morita Yoshihisa; Shieh Tien Tsan^*

来源：Journal of Differential Equations, 2018, 264(2): 550-574.

DOI：10.1016/j.jde.2017.09.015

• Reaction-diffusion system
• Mass conservation
• Equilibrium solution
• Asymptotic behavior
• Concentration phenomena
• Stability

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 k satisfying the relation epsilon := root g=root log k/k(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 K 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.

• 出版日期2018-1-15
• 单位中央民族大学