Upper Bounds for Coarsening for the Degenerate Cahn-hilliard Equation

The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 10], ut = ∇ · (1− u)∇ [Θ 2 {ln(1 + u)− ln(1− u)} − αu− 4u ] , is characterized by the growth of domains in which u(x, t) ≈ u±, where u± denote the ”equilibrium phases;” this process is known as coarsening. The degree of coarsening can be quantified in terms of a characteristic length scale, l(t), where l(t) is prescribed via a Liapunov functional and the (W 1,∞)∗ norm of u(x, t). In this paper, we prove upper bounds on l(t) for all temperatures Θ ∈ (0, Θc), where Θc denotes the ”critical temperature,” and for arbitrary mean concentrations, ū ∈ (u−, u+). Our results generalize the upper bounds obtained by Kohn & Otto [15]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.