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August  2007, 17(3): 541-552. doi: 10.3934/dcds.2007.17.541

Remarks on accessible steady states for some coagulation-fragmentation systems

Miguel A. Herrero 1, and  Marianito R. Rodrigo 2,

1. 

Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Plaza de las Ciencias, 28040 Madrid, Spain
2. 

Departamento Académico de Matemáticas, Instituto Tecnológico Autónomo de México, México D.F., Mexico

Received  January 2006 Revised  September 2006 Published  December 2006

In this paper we consider some systems of ordinary differential equations which are related to coagulation-fragmentation processes. In particular, we obtain explicit solutions $\{c_k(t)\}$ of such systems which involve certain coefficients obtained by solving a suitable algebraic recurrence relation. The coefficients are derived in two relevant cases: the high-functionality limit and the Flory-Stockmayer model. The solutions thus obtained are polydisperse (that is, $c_k(0)$ is different from zero for all $k \ge 1$) and may exhibit monotonically increasing or decreasing total mass. We also solve a monodisperse case (where $c_1(0)$ is different from zero but $c_k(0)$ is equal to zero for all $k \ge 2$) in the high-functionality limit. In contrast to the previous result, the corresponding solution is now shown to display a sol-gel transition when the total initial mass is larger than one, but not when such mass is less than or equal to one.
Keywords: exact solutions, sol-gel transition., coagulation-fragmentation, gelation.
Mathematics Subject Classification: Primary: 34A05; Secondary: 34A3.
Citation: Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541
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