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.