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

Jann Long Chern, Yoshihisa Morita, Tien Tsan Shieh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

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/κ2. 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.

Original languageEnglish
Pages (from-to)550-574
Number of pages25
JournalJournal of Differential Equations
Volume264
Issue number2
DOIs
Publication statusPublished - 2018 Jan 15
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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