Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability

Abstract

It is proved that the Cahn-Hilliard equation on a smooth domain possesses solutions which have spike layers localizing where the mean curvature of the boundary of the domain has nondegenerate critical points. Solutions of this type can be found with any average value which lies in the metastable region. It is also shown that these solutions have Morse indices at least equal to the number of spikes.