  The dynamics of a coagulation-fragmentation equation with multiplicative
coagulation kernel and critical singular fragmentation is studied. In contrast
to the coagulation equation, it is proved that fragmentation prevents the
occurrence of the gelation phenomenon and a mass-conserving solution is
constructed. The large time behavior of this solution is shown to be described
by a self-similar solution. In addition, the second moment is finite for
positive times whatever its initial value. The proof relies on the Laplace
transform which maps the original equation to a first-order nonlinear
hyperbolic equation with a singular source term. A precise study of this
equation is then performed with the method of characteristics.
