Dynamics of a Non-Autonomous ODE System Occurring in Coagulation Theory

We consider a constant coefficient coagulation equation with Becker–Döring type interactions and power law input of monomers J ₁(t) = α t ^ω, with α > 0 and $\omega > -\frac{1}{2}$. For this infinite dimensional system we prove solutions converge to similarity profiles as t and j converge to infinity in a similarity way, namely with either $j\!/\!\varsigma$ or $(j-\varsigma)/\sqrt{\varsigma}$ constants, where $\varsigma=\varsigma(t)$ is a function of t only. This work generalizes to the non-autonomous case a recent result of da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398. and provides a rigorous derivation of formal results obtained by Wattis J. Phys. A: Math. Gen. 37, 7823–7841. The main part of the approach is the analysis of a bidimensional non-autonomous system obtained through an appropriate change of variables; this is achieved by the use of differential inequalities and qualitative theory methods. The results about rate of convergence of solutions of the bidimensional system thus obtained are fed into an integral formula representation for the solutions of the infinite dimensional system which is then estimated by an adaptation of methods used by da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398.