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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 & 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.  doi: 10.1007/s10097-003-0055-3.  Google Scholar [2] J. Bertoin, Random Fragmentation and Coagulation Processes,, Cambridge Studies in Advanced Mathematics, (2006).  doi: 10.1017/CBO9780511617768.  Google Scholar [3] J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations,, preprint, ().   Google Scholar [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).  doi: 10.1088/0266-5611/30/2/025007.  Google Scholar [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.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar [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.  doi: 10.1080/17513750902935208.  Google Scholar [7] M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757.  doi: 10.1142/S021820251000443X.  Google Scholar [8] R. Drake, A general mathematical survey of the coagulation equation,, Topics in Current Aerosol Research (Part 2), (1972), 201.  doi: 10.1016/B978-0-08-016809-8.50003-6.  Google Scholar [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.  doi: 10.1016/j.anihpc.2004.06.001.  Google Scholar [10] A. G. Fredrickson, D. Ramkrishna and H. Tsuchiya, Statistics and dynamics of procaryotic cell populations,, Math. Biosci., 1 (1967), 327.  doi: 10.1016/0025-5564(67)90008-9.  Google Scholar [11] B. Haas, Loss of mass in deterministic and random fragmentations,, Stochastic Processes and their Applications, 106 (2003), 245.  doi: 10.1016/S0304-4149(03)00045-0.  Google Scholar [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.  doi: 10.1214/09-AAP622.  Google Scholar [13] E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable,, Springer-Verlag, (1965).  doi: 10.1007/978-3-662-29794-0.  Google Scholar [14] P. Michel, Existence of a solution to the cell division eigenproblem,, Math. Models Methods Appl. Sci., 16 (2006), 1125.  doi: 10.1142/S0218202506001480.  Google Scholar [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), (2015).  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar [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).  doi: 10.1186/1741-7007-12-17.  Google Scholar [17] J. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910.   Google Scholar

show all references

##### References:
 [1] J. Bertoin, The asymptotic behavior of fragmentation processes,, J. Eur. Math. Soc. (JEMS), 5 (2003), 395.  doi: 10.1007/s10097-003-0055-3.  Google Scholar [2] J. Bertoin, Random Fragmentation and Coagulation Processes,, Cambridge Studies in Advanced Mathematics, (2006).  doi: 10.1017/CBO9780511617768.  Google Scholar [3] J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations,, preprint, ().   Google Scholar [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).  doi: 10.1088/0266-5611/30/2/025007.  Google Scholar [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.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar [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.  doi: 10.1080/17513750902935208.  Google Scholar [7] M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757.  doi: 10.1142/S021820251000443X.  Google Scholar [8] R. Drake, A general mathematical survey of the coagulation equation,, Topics in Current Aerosol Research (Part 2), (1972), 201.  doi: 10.1016/B978-0-08-016809-8.50003-6.  Google Scholar [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.  doi: 10.1016/j.anihpc.2004.06.001.  Google Scholar [10] A. G. Fredrickson, D. Ramkrishna and H. Tsuchiya, Statistics and dynamics of procaryotic cell populations,, Math. Biosci., 1 (1967), 327.  doi: 10.1016/0025-5564(67)90008-9.  Google Scholar [11] B. Haas, Loss of mass in deterministic and random fragmentations,, Stochastic Processes and their Applications, 106 (2003), 245.  doi: 10.1016/S0304-4149(03)00045-0.  Google Scholar [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.  doi: 10.1214/09-AAP622.  Google Scholar [13] E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable,, Springer-Verlag, (1965).  doi: 10.1007/978-3-662-29794-0.  Google Scholar [14] P. Michel, Existence of a solution to the cell division eigenproblem,, Math. Models Methods Appl. Sci., 16 (2006), 1125.  doi: 10.1142/S0218202506001480.  Google Scholar [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), (2015).  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar [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).  doi: 10.1186/1741-7007-12-17.  Google Scholar [17] J. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910.   Google Scholar
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