We study the Neumann boundary value problem for stationary radial solutions of a quasilinear Cahn-Hilliard model in a ball in . We establish new results on the existence, uniqueness, and multiplicity (by "branching") of such solutions. We show striking differences in pattern formation produced by the Cahn-Hilliard model with the p-Laplacian and a potential () in place of the regular (linear) Laplace operator and a potential. The corresponding energy functional exhibits one-dimensional continua ("curves") of critical points as opposed to the classical case with the Laplace operator. These facts offer a different explanation of the "slow dynamics" on the attractor for the dynamical system generated by the corresponding time-dependent parabolic problem.
Published April 15, 2009.
Math Subject Classifications: 35J20, 35B45, 35P30, 46E35.
Key Words: Generalized Cahn-Hilliard and bi-stable equations; radial p-Laplacian; phase plane analysis; first integral; nonuniqueness for initial value problems.
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| Peter Takac |
Institut für Mathematik, Universität Rostock
D-18055 Rostock, Germany
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