Asymptotic behavior of solutions to the nonlinear breakage equations

The nonlinear breakage equations are a mathematical model for the dynamics of cluster growth when clusters undergo binary collisions resulting either in coalescence or breakup with possible transfer of matter. Each of these two events may happen with an a priori prescribed probability depending for instance on the sizes of the colliding clusters. The model consists of a countable number of non-locally coupled nonlinear ordinary differential equations modeling the concentration of the various clusters. In the present paper we consider asymptotic behavior of solutions as time tends to infinity and prove the weak* convergence to steady states provided at least two monoclusters appear after a collision, and the weak* convergence to some equilibrium state at the expense of making stronger assumptions. A result on the strong convergence has also been obtained.