• Previous Article
    Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
  • KRM Home
  • This Issue
  • Next Article
    Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry
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

[1]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[2]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[3]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[4]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[5]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[6]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[7]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[8]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[9]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[10]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[11]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[12]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[13]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[14]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[15]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[16]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[17]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[18]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[19]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[20]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]