# American Institute of Mathematical Sciences

June  2019, 8(2): 247-271. doi: 10.3934/eect.2019014

## On a Mathematical model with non-compact boundary conditions describing bacterial population (Ⅱ)

 LMCM-RSA, 22 Rue Des Canadiens, Poitiers, 86000, France

Received  June 2018 Revised  September 2018 Published  June 2019 Early access  March 2019

This work is a natural continuation of an earlier one [1] in which a mathematical model has been studied. This model is based on maturation-velocity structured bacterial population. The bacterial mitosis is mathematically described by a non-compact boundary condition. We investigate the spectral properties of the generated semigroup and we give an explicit estimation of the bound of its infinitesimal generator.

Citation: Mohamed Boulanouar. On a Mathematical model with non-compact boundary conditions describing bacterial population (Ⅱ). Evolution Equations and Control Theory, 2019, 8 (2) : 247-271. doi: 10.3934/eect.2019014
##### References:
 [1] M. Boulanouar, On a Mathematical model with non-compact boundary conditions describing bacterial population, Trans. Theory. and Stat. Physics, 42 (2013), 99-130.  doi: 10.1080/00411450.2013.866144. [2] M. Boulanouar, Transport equation for growing bacterial populations (Ⅰ), Electron. J. Diff. Equ., 221 (2012), 1-25. [3] M. Boulanouar, Un modèle de Rotenberg avec la loi à mémoire parfaite, C.R.A.S. Paris Série I Math., 327 (1998), 965-968.  doi: 10.1016/S0764-4442(99)80161-X. [4] M. Boulanouar, On a Mathematical model with non-compact boundary conditions describing bacterial population : Asynchronous Growth Property, Submitted [5] C. V. M. van der Mee and P. Zweifel, A Fokker-Planck equation for growing cell populations, J. Math. Biol., 25 (1987), 61-72.  doi: 10.1007/BF00275888. [6] C. V. M. van der Mee, A transport equation modeling in cell growth, Stochastic Modeling in Biology (P. Tautu, Eds), Word Sci., Publishing, 1990,381–398. [7] W. Desch, I. Lasiecka and W. Schappacher, Feedback boundary control problems for linear semigroups, Isr., J., Math., 51 91985), 177–207. doi: 10.1007/BF02772664. [8] B. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149-153.  doi: 10.1007/BF01162028. [9] M. Rotenberg, Transport theory for growing cell populations, J. Theor. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.

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##### References:
 [1] M. Boulanouar, On a Mathematical model with non-compact boundary conditions describing bacterial population, Trans. Theory. and Stat. Physics, 42 (2013), 99-130.  doi: 10.1080/00411450.2013.866144. [2] M. Boulanouar, Transport equation for growing bacterial populations (Ⅰ), Electron. J. Diff. Equ., 221 (2012), 1-25. [3] M. Boulanouar, Un modèle de Rotenberg avec la loi à mémoire parfaite, C.R.A.S. Paris Série I Math., 327 (1998), 965-968.  doi: 10.1016/S0764-4442(99)80161-X. [4] M. Boulanouar, On a Mathematical model with non-compact boundary conditions describing bacterial population : Asynchronous Growth Property, Submitted [5] C. V. M. van der Mee and P. Zweifel, A Fokker-Planck equation for growing cell populations, J. Math. Biol., 25 (1987), 61-72.  doi: 10.1007/BF00275888. [6] C. V. M. van der Mee, A transport equation modeling in cell growth, Stochastic Modeling in Biology (P. Tautu, Eds), Word Sci., Publishing, 1990,381–398. [7] W. Desch, I. Lasiecka and W. Schappacher, Feedback boundary control problems for linear semigroups, Isr., J., Math., 51 91985), 177–207. doi: 10.1007/BF02772664. [8] B. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149-153.  doi: 10.1007/BF01162028. [9] M. Rotenberg, Transport theory for growing cell populations, J. Theor. Biol., 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.
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