Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 227-254.
Title: Stationary radial solutions for a quasilinear
Cahn-Hilliard model in N space dimensions
Author: Peter Takac (Univ. Rostock, Rostock, Germany)
Abstract:
We study the Neumann boundary value problem for stationary radial
solutions of a quasilinear Cahn-Hilliard model in a ball
$B_R(\mathbf{0})$ in $\mathbb{R}^N$. 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
$C^{1,\alpha}$ potential ($0<\alpha\leq 1$) in place of the
regular (linear) Laplace operator and a $C^2$ 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.