Uniqueness of solutions to the coagulation–fragmentation equation with singular kernel

Abstract

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space $\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)$, where $\varOmega _{\lambda , r_2} (T)$ is the space of all continuous functions f such that

$$\begin{aligned} \Vert f\Vert _{\lambda , r_2} : = \sup \limits _{0\le t \le T} \int _0^{\infty } \left( \exp (\lambda x ) + \frac{1}{x^{r_2}}\right) |f(x,t)| dx ~<~ \infty \end{aligned}$$

and $0< r_2 < 1$.