# American Institute of Mathematical Sciences

February  2009, 25(1): 251-272. doi: 10.3934/dcds.2009.25.251

## Upper bounds for coarsening for the degenerate Cahn-Hilliard equation

 1 Department of Mathematics, Technion-IIT, Haifa 32000 2 Institute of Applied Mathematics and Mechanics, 83114 Donetsk

Received  December 2007 Revised  October 2008 Published  June 2009

The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 9],

$u_t=\nabla \cdot (1-u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)-\ln(1-u)\} - \alpha u -$ Δu$],$

is characterized by the growth of domains in which $u(x,t) \approx u_{\pm},$ where $u_\pm$ 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 a $W^{1, \infty}$ predual norm of $u(x,t).$ In this paper, we prove upper bounds on $l(t)$ for all temperatures $\Theta \in (0, \Theta_c),$ where $\Theta_c$ denotes the ''critical temperature," and for arbitrary mean concentrations, $\bar{u}\in (u_{-}, u_{+}).$ Our results generalize the upper bounds obtained by Kohn & Otto [14]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.

Citation: Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
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