On the convergence to the smooth self-similar solution in the LSW model

We investigate the long-time behavior of solutions to the classical mean-field model by Lifshitz-Slyozov and Wagner (LSW). In the original work [4, 10] convergence of solutions to a uniquely determined self-similar solution was predicted. However, it is by now well known [2,5,7] that the long-time behavior of solutions depends sensitively on the initial data. In [5, 7, 8] a necessary and sufficient criterion for convergence to any self-similar solution which behaves like a finite power at the end of its (compact) support is given. In this paper we establish corresponding results for the LSW-solution which decays faster than any power. It turns out that the respective criterion for convergence to self-similarity is much less stringent than for the case of non-smooth self-similar solutions.