  Surface growth models may give rise to unstable growth with mound formation
whose tipical linear size L increases in time. In one dimensional systems
coarsening is generally driven by an attractive interaction between domain
walls or kinks. This picture applies to growth models for which the largest
surface slope remains constant in time (model B): coarsening is known to be
logarithmic in the absence of noise (L(t)=log t) and to follow a power law
(L(t)=t^{1/3}) when noise is present. If surface slope increases indefinitely,
the deterministic equation looks like a modified Cahn-Hilliard equation: here
we study the late stage of coarsening through a linear stability analysis of
the stationary periodic configurations and through a direct numerical
integration. Analytical and numerical results agree with regard to the
conclusion that steepening of mounds makes deterministic coarsening faster: if
alpha is the exponent describing the steepening of the maximal slope M of
mounds (M^alpha = L) we find that L(t)=t^n: n is equal to 1/4 for 1<alpha<2 and
it decreases from 1/4 to 1/5 for alpha>2, according to n=alpha/(5*alpha -2). On
the other side, the numerical solution of the corresponding stochastic equation
clearly shows that in the presence of shot noise steepening of mounds makes
coarsening slower than in model B: L(t)=t^{1/4}, irrespectively of alpha.
Finally, the presence of a symmetry breaking term is shown not to modify the
coarsening law of model alpha=1, both in the absence and in the presence of
noise.
