Asymptotic and effective coarsening exponents in surface growth models

We consider a class of unstable surface growth models, $\partial_t z = -\partial_x {\cal J}$, developing a mound structure of size $\lambda$ and displaying a perpetual coarsening process, i.e. an endless increase in time of $\lambda$. The coarsening exponents n, defined by the growth law of the mound size $\lambda$ with time, $\lambda \sim t^n$, were previously found by numerical integration of the growth equations [A. Torcini, P. Politi, Eur. Phys. J. B 25, 519 (2002)]. Recent analytical work now allows to interpret such findings as finite time effective exponents. The asymptotic exponents are shown to appear at so large time that cannot be reached by direct integration of the growth equations. The reason for the appearance of effective exponents is clearly identified.