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

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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 languageEnglish
Pages (from-to)283-302
Number of pages20
JournalQuarterly of Applied Mathematics
Volume58
Issue number2
Publication statusPublished - Jun 2000
Externally publishedYes

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ASJC Scopus subject areas

  • Applied Mathematics

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