TY - JOUR

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

AU - Taniguchi, M.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

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

U2 - 10.1090/qam/1753400

DO - 10.1090/qam/1753400

M3 - Article

AN - SCOPUS:0033730902

VL - 58

SP - 283

EP - 302

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 2

ER -