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June  2016, 9(2): 251-297. doi: 10.3934/krm.2016.9.251

Time asymptotics for a critical case in fragmentation and growth-fragmentation equations

1. 

Sorbonne Universités, Inria, UPMC Univ Paris 06, Lab. J.-L. Lions, UMR CNRS 7598, 75005 Paris, France

2. 

Departamento de Matemáticas, Universidad del País Vasco, (UPV/EHU), Apartado 644, E-48080 Bilbao

Received  January 2015 Revised  August 2015 Published  March 2016

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to the description of the long-time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation: $$\frac{\partial }{\partial t} u(t,x) + u(t,x) = \int\limits_x^\infty k_0 (\frac{x}{y}) u(t,y) dy.$$ Using the Mellin transform of the equation, we determine the long-time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.
Citation: Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic and Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251
References:
[1]

J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc. (JEMS), 5 (2003), 395-416. doi: 10.1007/s10097-003-0055-3.

[2]

J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics, 102, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617768.

[3]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, preprint, arXiv:1506.09187.

[4]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007, 28 pp. doi: 10.1088/0266-5611/30/2/025007.

[5]

M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathèmatiques Pures et Appliquèes, 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003.

[6]

V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamic with size dependency - strain phenomena, J. of Biol. Dyn., 4 (2010), 28-42. doi: 10.1080/17513750902935208.

[7]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[8]

R. Drake, A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), (1972), 201-376. doi: 10.1016/B978-0-08-016809-8.50003-6.

[9]

M. Escobedo, S. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001.

[10]

A. G. Fredrickson, D. Ramkrishna and H. Tsuchiya, Statistics and dynamics of procaryotic cell populations, Math. Biosci., 1 (1967), 327-374. doi: 10.1016/0025-5564(67)90008-9.

[11]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Processes and their Applications, 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0.

[12]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429. doi: 10.1214/09-AAP622.

[13]

E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer-Verlag, New York, 1965. doi: 10.1007/978-3-662-29794-0.

[14]

P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153. doi: 10.1142/S0218202506001480.

[15]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis (2015), in press. doi: 10.1016/j.anihpc.2015.01.007.

[16]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division in escherichia coli is triggered by a size-sensing rather than a timing mechanism, BMC Biology, 12 (2014), p17. doi: 10.1186/1741-7007-12-17.

[17]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.

show all references

References:
[1]

J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc. (JEMS), 5 (2003), 395-416. doi: 10.1007/s10097-003-0055-3.

[2]

J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics, 102, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617768.

[3]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, preprint, arXiv:1506.09187.

[4]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007, 28 pp. doi: 10.1088/0266-5611/30/2/025007.

[5]

M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathèmatiques Pures et Appliquèes, 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003.

[6]

V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamic with size dependency - strain phenomena, J. of Biol. Dyn., 4 (2010), 28-42. doi: 10.1080/17513750902935208.

[7]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[8]

R. Drake, A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), (1972), 201-376. doi: 10.1016/B978-0-08-016809-8.50003-6.

[9]

M. Escobedo, S. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001.

[10]

A. G. Fredrickson, D. Ramkrishna and H. Tsuchiya, Statistics and dynamics of procaryotic cell populations, Math. Biosci., 1 (1967), 327-374. doi: 10.1016/0025-5564(67)90008-9.

[11]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Processes and their Applications, 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0.

[12]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429. doi: 10.1214/09-AAP622.

[13]

E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer-Verlag, New York, 1965. doi: 10.1007/978-3-662-29794-0.

[14]

P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153. doi: 10.1142/S0218202506001480.

[15]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis (2015), in press. doi: 10.1016/j.anihpc.2015.01.007.

[16]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division in escherichia coli is triggered by a size-sensing rather than a timing mechanism, BMC Biology, 12 (2014), p17. doi: 10.1186/1741-7007-12-17.

[17]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.

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