## 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=logκ/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.

Original language | English |
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Pages (from-to) | 550-574 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 264 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Jan 15 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics