The discrete diffusive coagulation-fragmentation equations with scattering
The author considers a model of coagulation-fragmentation with diffusion in which the coagulation clusters are assumed to be multiples of an elementary identical unit. Moreover, the model is endowed with a suitable scattering mechanism that prevents from exceeding a maximal cluster size, see \textit{A. Fasano} and \textit{F. Rosso} [Lectures on applied mathematics, Proceedings of the symposium organized by the Sonderforschungsbereich 438 on the occasion of Karl-Heinz Hoffmann's 60th birthday, 123--141 (2000; Zbl 1049.82525)]. The state of the system at time \(t\) is thus described by \(u_i = u_i(x,t)\), \(i = 1,2,\dots,M\), where \(x \in \Omega \subset {\mathbb R}^n\) is the position variable and \(i\) is the cluster size. The set of functions \(u = (u_1,u_2,\dots,u_n)\) is assumed to satisfy a set of reaction-diffusion equations of the following form: \[ \dot u_i - d_i \Delta u_i = f_i(x,u), \quad i = 1,2,\dots,M, \] with assigned initial and Neumann boundary conditions. The functions \(f_i\) account for (mass-preserving) coagulation-fragmentation and scattering. The problem is recast into a semilinear Cauchy problem in a suitable \(L^p\)-like space, with \(2 \leq p < \infty\) and \(2p > n\). Local-in-time existence and uniqueness of mild and strong solutions is proved under fairly general assumptions. Global-in-time existence and uniqueness is proved under the assumption \(d_1 = d_2 = \cdots =d_M\). Under the same assumption, in the last part of the paper a careful analysis of the asymptotic behaviour for \(t \to +\infty\) is carried on, focusing on the three special (but physically relevant) cases of dominating coagulation, dominating fragmentation, and detailed-balance condition.
scientific article; zbMATH DE number 2092996