June  2012, 5(2): 223-236. doi: 10.3934/krm.2012.5.223

The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth

1. 

School of Mathematical Sciences, UKZN, Durban, South Africa

2. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow, Scotland

Received  August 2011 Revised  November 2011 Published  April 2012

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible.
Citation: Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223
References:
[1]

W. Arendt and A. Rhandi, Perturbation of positive semigroups, Arch. Math. (Basel), 56 (1991), 107-119. doi: 10.1007/BF01200341.  Google Scholar

[2]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Stat. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961.  Google Scholar

[3]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[4]

J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89.  Google Scholar

[5]

J. Banasiak, On an irregular dynamics of certain fragmentation semigroups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 361-377. doi: 10.1007/s13398-011-0015-9.  Google Scholar

[6]

J. Banasiak, Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates, Nonlinear Anal. Real World Appl., 13 (2012), 91-105. doi: 10.1016/j.nonrwa.2011.07.016.  Google Scholar

[7]

J. Carr and F. P. da Costa, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Stat. Phys., 77 (1994), 89-123. doi: 10.1007/BF02186834.  Google Scholar

[8]

Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, "One-Parameter Semigroups," CWI Monographs, 5, North Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[9]

F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, J. Math. Anal. Appl., 192 (1995), 892-914. doi: 10.1006/jmaa.1995.1210.  Google Scholar

[10]

K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[11]

K.-J. Engel and R. Nagel, "A Short Course on One Parameter Semigroups," Springer, New York, 2005.  Google Scholar

[12]

P. Glendinning, "Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1994.  Google Scholar

[13]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[14]

P. Laurençot, The discrete coagulation equations with multiple fragmentation, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67-82.  Google Scholar

[15]

A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation, Physica D, 239 (2010), 1436-1445. doi: 10.1016/j.physd.2009.03.013.  Google Scholar

[16]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.  Google Scholar

[17]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Pol. Ac. Sci. Math., 43 (1995), 245-262.  Google Scholar

[18]

A. L. Smith, W. Lamb, M. Langer and A. C. McBride, Discrete fragmentation with mass loss, J. Evol. Equ., 12 (2012), 181-201. doi: 10.1007/s00028-011-0129-8.  Google Scholar

show all references

References:
[1]

W. Arendt and A. Rhandi, Perturbation of positive semigroups, Arch. Math. (Basel), 56 (1991), 107-119. doi: 10.1007/BF01200341.  Google Scholar

[2]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Stat. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961.  Google Scholar

[3]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[4]

J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89.  Google Scholar

[5]

J. Banasiak, On an irregular dynamics of certain fragmentation semigroups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 361-377. doi: 10.1007/s13398-011-0015-9.  Google Scholar

[6]

J. Banasiak, Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates, Nonlinear Anal. Real World Appl., 13 (2012), 91-105. doi: 10.1016/j.nonrwa.2011.07.016.  Google Scholar

[7]

J. Carr and F. P. da Costa, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Stat. Phys., 77 (1994), 89-123. doi: 10.1007/BF02186834.  Google Scholar

[8]

Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, "One-Parameter Semigroups," CWI Monographs, 5, North Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[9]

F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, J. Math. Anal. Appl., 192 (1995), 892-914. doi: 10.1006/jmaa.1995.1210.  Google Scholar

[10]

K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[11]

K.-J. Engel and R. Nagel, "A Short Course on One Parameter Semigroups," Springer, New York, 2005.  Google Scholar

[12]

P. Glendinning, "Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1994.  Google Scholar

[13]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[14]

P. Laurençot, The discrete coagulation equations with multiple fragmentation, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67-82.  Google Scholar

[15]

A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation, Physica D, 239 (2010), 1436-1445. doi: 10.1016/j.physd.2009.03.013.  Google Scholar

[16]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.  Google Scholar

[17]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Pol. Ac. Sci. Math., 43 (1995), 245-262.  Google Scholar

[18]

A. L. Smith, W. Lamb, M. Langer and A. C. McBride, Discrete fragmentation with mass loss, J. Evol. Equ., 12 (2012), 181-201. doi: 10.1007/s00028-011-0129-8.  Google Scholar

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