On a Class of Continuous Collision-Induced Breakage Equation

Abstract

In this work, we establish the existence of mass-conserving weak solutions to a nonlinear collision-induced breakage equation in which binary collisions may trigger particle breakup. The result is proved for a class of product-type collision kernels whose small-size behavior is controlled by a power-law function of the form $ω_0(x)\le A_1\,x^\ell$, while no growth restriction is imposed on the large-size factor $ω_\infty$. The qualitative behavior of the solutions depends crucially on the exponent $\ell$ near the origin. Sublinear growth corresponding to $\ell<\tfrac12$ yields existence only on finite time intervals, whereas superlinear growth corresponding to $\ell>\tfrac12$ ensures global-in-time existence.