Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability. (English) Zbl 1157.35407

Bates, Peter W.; Dancer, E. Norman; Shi, Junping

Adv. Differ. Equ. 4, No. 1, 1-69 (1999).

Summary: 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.

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B25	Singular perturbations in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
58E50	Applications of variational problems in infinite-dimensional spaces to the sciences