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

Remarks on accessible steady states for some coagulation-fragmentation systems

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.
Citation: Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541
[1]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043

[2]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014

[3]

Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009

[4]

Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040

[5]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

[6]

Jacek Banasiak. Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020161

[7]

Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589

[8]

Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445

[9]

Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020012

[10]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563

[11]

Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177

[12]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002

[13]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004

[14]

Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007

[15]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[16]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011

[17]

Kota Kumazaki. Periodic solutions for non-isothermal phase transition models. Conference Publications, 2011, 2011 (Special) : 891-902. doi: 10.3934/proc.2011.2011.891

[18]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[19]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125

[20]

Tong Zhang, Yuxi Zheng. Exact spiral solutions of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 117-133. doi: 10.3934/dcds.1997.3.117

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]