RESUMEN
This paper gives theoretical
results on spinodal decomposition for the Cahn--Hillard equation. We prove a
mechanism which explains why most solutions for the Cahn--Hilliard equation
starting near a homogeneous equilibrium within the spinodal interval exhibit
phase separation with a characteristic wavelength when exiting a ball of radius
R. Namely, most solutions are driven into a region of phase space in which linear
behavior dominates for much longer than expected.
The Cahn--Hilliard equation
depends on a small parameter $\epsilon,$ modeling the
(atomic scale) interaction length; we quantify the behavior of solutions as
$\epsilon \rightarrow 0$. Specifically, we show that most solutions starting
close to the homogeneous equilibrium remain close to the corresponding solution
of the linearized equation with relative distance $O(\epsilon^{2-n/2})$ up to a
ball of radius R in the $H^2(\Omega)$-norm, where R is proportional to
$\epsilon^{-1+\rho+n/4}$ as $\epsilon \to 0$. Here, $n \in \{
1,2,3 \}$ denotes the dimension of the considered domain, and $\rho > 0$ can
be chosen arbitrarily small. Not only does this approach significantly increase
the radius of explanation for spinodal decomposition, but it also gives a clear
picture of how the phenomenon occurs.
While these results hold for
the standard cubic nonlinearity, we also show that considerably better results
can be obtained for similar higher order nonlinearities. In particular, we
obtain $R \sim \epsilon^{-2 + \rho + n/2}$ for every
$\rho > 0$ by choosing a suitable nonlinearity.
