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Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions
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 |
$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.
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