  In this paper we derive optimal algebraic-in-time relaxation rates for the
Cahn-Hilliard equation on the one-dimensional torus and the line. We consider
initial conditions with a finite (but not small) $L^1$-distance to an
appropriately defined bump. The result extends the relaxation method developed
previously for a single transition layer (the "kink") to the case of two
transition layers (the "bump"). As in the previous work, the tools include
Nash-type inequalities, duality arguments, and Schauder estimates. For both the
kink and the bump, the energy gap is translation invariant and its decay alone
cannot specify to which member of the family of minimizers the solution
converges. Whereas in the case of the kink, the conserved quantity singles out
the longtime limit, in the case of a bump, a new argument is needed. On the
torus, we quantify convergence to the bump that is the longtime limit; on the
line, the bump-like states are merely metastable and we quantify the initial
relaxation behavior.
