Mass conserving global solutions for the nonlinear collision-induced
fragmentation model with a singular kernel
Abstract
This article is devoted to the study of existence of a mass conserving
global solution for the collision-induced nonlinear fragmentation model
which arises in particulate processes, with the following type of
collision kernel:
\[C(x,y)~\le~k_1
\frac{(1 + x)^\nu (1 +
y)^\nu}{\left(xy\right)^\sigma},\]
for all ~$x, y \in
(0,\infty)$, where $k_1$ is a positive constant,
$\sigma \in
\left[0,\tfrac{1}{2}\right]$
and $\nu \in [0, 1]$. The
above-mentioned form includes many practical oriented kernels of both
\emph{singular} and
\emph{non-singular} types. The singularity of the
unbounded collision kernel at coordinate axes extends the previous
existence result of Paul and Kumar [Mathematical Methods in the
Applied Sciences 41 (7) (2018) 2715–2732
(\href{https://doi.org/10.1002/mma.4775}{doi:10.1002/mma.4775})]
and also exhibits at most quadratic growth at infinity. Finally,
uniqueness of solution is also investigated for pure singular collision
rate, i.e., for ~ $\nu=0$.