Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation

We study the asymptotic behaviour of classes of global and blow-up solutions of a semilinear parabolic equation of Cahn-Hilliard type ut = −∆(∆u + |u|u) in R ×R+, p > 1, with bounded integrable initial data. We show that in some {p, N}-parameter ranges it admits a countable set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exponent p = p0 = 1 + 2 N , where the first simplest profile is shown to be stable. Unlike the blow-up case, we show that, for p = p0, the set of global decaying source-type similarity solutions is continuous and determine the stable mass-branch. We prove that there exists a countable spectrum of critical exponents {p = pl = 1+ 2 N+l , l = 0, 1, 2, ...} creating bifurcation branches, which play a key role in general description of solutions globally decaying as t → ∞.