A formula for the post-gelation mass of a coagulation equation with a separable bilinear kernel (Q852890)

The authors consider the coagulation equation \[ \dfrac{\partial c}{\partial t}(\lambda, t)=\dfrac{1}{2}\int^{\lambda}_0 K(\lambda-\mu, \mu)c(\lambda-\mu, t)c(\mu,t)\,d\mu-c(\lambda,t)\int^\infty_0 K(\lambda, \mu)c(\mu,t)\,d\mu\tag{1} \] \( c(\lambda, 0)=c_0(\lambda)\) with the specific choice of the kernel \[ K(\lambda, \mu)=\theta(\lambda)\theta(\mu),\quad\theta(\lambda)=\alpha +\beta\lambda,\quad \alpha, \beta\geq 0. \] Their attention is focussed on the so-called gelation phenomenon, corresponding to breakdown of mass conservation, due to the formation of a ``superparticle'', which necessarily occurs if \(\beta>0\). In particular they want to describe the behaviour of the system after gelation. Using Laplace transform equation (1) is reduced to a nonlinear first order PDE, which however is of non-standard type, since it includes the unknown value of the solution along one of the characteristics of the equation. The authors manage to provide the solution of this problem, obtaining the explicit expression of gelation time, depending on the initial data. Then they proceed further, deriving the time evolution of mass \(M(t)\) after gelation by means of an elegant method. Examples for various choices of initial mass distributions are illustrated. the asymptotic behaviour of \(M(t)\) for large \(t\) is also discussed.