The LSW model with encounters has been suggested by Lifshitz and
Slyozov as a regularization of their classical mean-field model for
domain coarsening to obtain universal self-similar long-time
behavior. We rigorously establish that an exponentially decaying
self-similar solution to this model exist, and show that this
solutions is isolated in a certain function space. Our proof relies
on setting up a suitable fixed-point problem in an appropriate
function space and careful asymptotic estimates of the solution to a
corresponding homogeneous problem.
Recently, Dreyer and Duderstadt have proposed a modification of the
Becker-Doering cluster equations which now have a nonconvex
Lyapunov function. We start with existence and uniqueness results
for the modified equations. Next we derive an explicit criterion for
the existence of equilibrium states and solve the minimization
problem for the Lyapunov function. Finally, we discuss the long time
behavior in the case that equilibrium solutions do exist.