Unexpectedly linear behavior for the Cahn-Hilliard equation

This paper gives theoretical results on spinodal decomposition for the Cahn -Hillard equation. We prove a mechanism which explains why most solutions f or the Cahn-Hilliard equation starting near a homogeneous equilibrium withi n the spinodal interval exhibit phase separation with a characteristic wave length when exiting a ball of radius R. Namely, most solutions are driven i nto a region of phase space in which linear behavior dominates for much lon ger than expected. The Cahn-Hilliard equation depends on a small parameter epsilon, modeling t he (atomic scale) interaction length; we quantify the behavior of solutions as epsilon --> 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 epsilo n(-1+rho+n/4) as epsilon --> 0. Here, n is an element of {1, 2, 3} denotes the dimension of the considered domain, and rho > 0 can be chosen arbitrari ly small. Not only does this approach significantly increase the radius of explanation for spinodal decomposition, but it also gives a clear picture o f 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 n onlinearities. In particular, we obtain R similar to epsilon(-2+rho+n/2) fo r every rho > 0 by choosing a suitable nonlinearity.