Symmetry breaking bifurcation and the stability of stationary solutions of a phase-field model

We have been investigating the global bifurcation diagrams of stationary solutions for a phase field model proposed by Fix and Caginalp in a one-dimensional case. It has recently been shown that there exists a secondary bifurcation with a symmetry-breaking phenomenon from a branch consisting of symmetric solutions in the case where the total enthalpy equals zero. In this paper, we determine the stability/instability of all symmetric solutions and asymmetric solutions near the secondary bifurcation point. Moreover, we show representation formulas for all eigenvalues and eigenfunctions for the linearized eigenvalue problem around the symmetric solutions.