Existence and non-existence for the collision-induced breakage equation

A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^{alpha} y^{beta} + x^{beta} y^{alpha}$ with $alpha le beta le 1$. When $alpha + beta in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $alpha + beta in [0,1)$ and $alpha ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $alpha <0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.