Mass-conserving weak solutions to the continuous nonlinear fragmentation equation in the presence of mass transfer

Abstract

A mathematical model for the continuous nonlinear fragmentation equation is considered in the presence of mass transfer. In this paper, we demonstrate the existence of mass-conserving weak solutions to the nonlinear fragmentation equation with mass transfer for collision kernels of the form $\Phi (x,y)=\kappa (x^{{\sigma }_1}{y}^{{\sigma }_2}+y^{{\sigma }_1}{x}^{{\sigma }_2})$, $\kappa >0$, $0\le {\sigma }_1\le {\sigma }_2\le 1$, and ${\sigma }_1\ne 1$ for $(x,y)\in R_{+}^2$, with integrable daughter distribution functions, thereby extending previous results obtained by Giri & Laurençot (2021). In particular, the existence of at least one global weak solution is shown when the collision kernel exhibits at least linear growth, and one local weak solution when the collision kernel exhibits sublinear growth. In both cases, finite superlinear moment bounds are obtained for positive times without requiring the finiteness of initial superlinear moments. Additionally, the uniqueness of solutions is confirmed in both cases.