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

Taniguchi, M.

doi:10.1090/qam/1753400

Author: M. Taniguchi
Journal: Quart. Appl. Math. 58 (2000), 283-302
MSC: Primary 35R35; Secondary 34B15, 35B35
DOI: https://doi.org/10.1090/qam/1753400
MathSciNet review: MR1753400
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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.

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