Existence, stability and nonlinear estimates of stationary-state solutions to the nonlinear aggregation with collision-induced fragmentation model

Abstract

Existence and uniqueness of a stationary-state solution to the nonlinear aggregation and collision-induced fragmentation equation is proved over a weighted $L^1$-space. The assumption of a detailed balance condition is relaxed to attain the existence of the solution. Aggregation and fragmentation kernels are considered to exhibit linear and quadratic growth rates respectively which encompass a wide range of physically significant kernels. Asymptotic properties of the time-dependent solution are analyzed in detail and convergence of the same to the stationary-state solution is also examined. Exponential rate of convergence is obtained by proving the asymptotic stability of the stationary-state solution. Further, nonlinear estimates of the solution are obtained using semigroup theory of operators. The study is further extended to analyze the nonexistence of a stationary-state solution for a particular choice of kinetic kernels over a suitably constructed solution space. A numerical example is provided in order to visualize the nonexistence of a stationary-state solution and other physical quantities.