  The discrete collisional breakage equation, which captures the dynamics of
cluster growth when clusters encounter binary collisions with possible matter
transfer, is discussed in this article. The existence of global mass-conserving
solutions is investigated for the collision kernels $a_{i,j}=A(i^{\alpha}
j^{\beta} + i^{\beta}j^{\alpha})$, $i, j \ge 1$, with $\alpha \in (-\infty,1)$,
$\beta\in [\alpha,1]\cap (0,1]$, and $A>0$ and for a large class of possibly
unbounded daughter distribution functions. All algebraic superlinear moments of
these solutions are bounded on time intervals $[T,\infty)$ for any $T>0$. The
uniqueness issue is further handled under additional restrictions on the
initial data. Finally, non-trivial stationary solutions are constructed by a
dynamical approach.
