### Abstract

This paper studies construction and linear stability of spherical interfaces in an equilibrium state in a two-phase boundary problem arising in activator-inhibitor models in chemistry. By studying the linearized eigenvalue problem near a given equilibrium ball, we show that the eigenvalues with nonnegative real parts are all real, and that they are characterized as values of a strictly convex function for specific discrete values of its argument. The stability is determined by the location of the zero points of this convex function. Using this fact, we present a criterion of stability in a useful form. We show examples and illustrate that stable equilibrium balls and unstable ones coexist near saddle-node bifurcation points in the bifurcation diagram, and a given equilibrium ball located far from bifurcation points is unstable and the eigenfunction associated with the largest eigenvalue consists of spherically harmonic functions of high degrees.

Original language | English |
---|---|

Pages (from-to) | 283-302 |

Number of pages | 20 |

Journal | Quarterly of Applied Mathematics |

Volume | 58 |

Issue number | 2 |

Publication status | Published - Jun 2000 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem.** / Taniguchi, Masaharu.

Research output: Contribution to journal › Article

*Quarterly of Applied Mathematics*, vol. 58, no. 2, pp. 283-302.

}

TY - JOUR

T1 - Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem

AU - Taniguchi, Masaharu

PY - 2000/6

Y1 - 2000/6

N2 - This paper studies construction and linear stability of spherical interfaces in an equilibrium state in a two-phase boundary problem arising in activator-inhibitor models in chemistry. By studying the linearized eigenvalue problem near a given equilibrium ball, we show that the eigenvalues with nonnegative real parts are all real, and that they are characterized as values of a strictly convex function for specific discrete values of its argument. The stability is determined by the location of the zero points of this convex function. Using this fact, we present a criterion of stability in a useful form. We show examples and illustrate that stable equilibrium balls and unstable ones coexist near saddle-node bifurcation points in the bifurcation diagram, and a given equilibrium ball located far from bifurcation points is unstable and the eigenfunction associated with the largest eigenvalue consists of spherically harmonic functions of high degrees.

AB - This paper studies construction and linear stability of spherical interfaces in an equilibrium state in a two-phase boundary problem arising in activator-inhibitor models in chemistry. By studying the linearized eigenvalue problem near a given equilibrium ball, we show that the eigenvalues with nonnegative real parts are all real, and that they are characterized as values of a strictly convex function for specific discrete values of its argument. The stability is determined by the location of the zero points of this convex function. Using this fact, we present a criterion of stability in a useful form. We show examples and illustrate that stable equilibrium balls and unstable ones coexist near saddle-node bifurcation points in the bifurcation diagram, and a given equilibrium ball located far from bifurcation points is unstable and the eigenfunction associated with the largest eigenvalue consists of spherically harmonic functions of high degrees.

UR - http://www.scopus.com/inward/record.url?scp=0033730902&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033730902&partnerID=8YFLogxK

M3 - Article

VL - 58

SP - 283

EP - 302

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 2

ER -