  We formulate dynamical rate equations for physical processes driven by a
combination of diffusive growth, size fragmentation and fragment coagulation.
Initially, we consider processes where coagulation is absent. In this case we
solve the rate equation exactly leading to size distributions of Bessel type
which fall off as $\exp(-x^{3/2})$ for large $x$-values. Moreover, we provide
explicit formulas for the expansion coefficients in terms of Airy functions.
Introducing the coagulation term, the full non-linear model is mapped exactly
onto a Riccati equation that enables us to derive various asymptotic solutions
for the distribution function. In particular, we find a standard exponential
decay, $\exp(-x)$, for large $x$, and observe a crossover from the Bessel
function for intermediate values of $x$. These findings are checked by
numerical simulations and we find perfect agreement between the theoretical
predictions and numerical results.
