Similarity solutions of a Becker-D¨ oring system with time-dependent monomer input Jonathan AD Wattis Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K. E-mail: Jonathan.Wattis@nottingham.ac.uk Abstract. We formulate the Becker-D¨ oring equations for cluster growth in the presence of a time-dependent source of monomer input. In the case of size-independent aggregation and fragmentation rate coefficients we find similarity solutions which are approached in the large time limit. The form of the solutions depends on the rate of monomer input and whether fragmentation is present in the model; four distinct types of solution are found. PACS numbers: 64.60.-i, 82.60.Nh, 05.70.Fh, aggregation equations, coagulation- fragmentation equations, self-similar behaviour. 1. Introduction Until recently, self-similar behaviour in the Becker-D¨ oring equations has been almost unknown. However self-similar behaviour in the Smoluchowski coagulation equations [24] is commonplace and widely studied, for example, see Leyvraz [17] for a recent review. In the pure aggregation formulation of the Smoluchowski model [24] dc r dt = 1 2 r−1  s=1 a s,r−s c s c r−s − ∞  s=1 a r,s c r c s , (1.1) attraction to self-similar solutions are observed in the explicitly solveable cases a r,s = a and a r,s = ars as well as many other kernels. The former has the form c r (t)= t −2 ψ(r/t) with ψ(η) = (4/a 2 )e −2aη/ ; existence and convergence results have been proved for various cases by Kreer [16], da Costa [7] and Menon & Pego [21, 22]. The gelling solution for the kernel a r,s = ars was first found by Leyvraz & Tschudi [18]. For arbitrary initial data, this system undergoes a gelation transition at t g =1/a ∑ ∞ r=1 r 2 c r (0); for t>t g the solution has the form c r (t)= t −1 φ r with φ r independent of time, which is formally a similarity solution, and attracting for all initial data. Krapivsky & Redner [15] have studied the differences in asymptotic structure between constant monomer and constant mass formulations of aggregation problems. Their study, however, was on the Smoluchowski coagulation equations, which allows