# American Institute of Mathematical Sciences

January  2011, 29(1): 305-321. doi: 10.3934/dcds.2011.29.305

## Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions

 1 Clermont Université, Université Blaise Pascal, Laboratoire de Matématiques, BP. 10448, F-63000 CLERMONT-FERRAND, CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 AUBIERE, France 2 Département de Mathématiques, Faculté des Sciences de GABÈS, Cité Erriadh 6072, Zrig, GABÈS, Tunisia

Received  July 2009 Revised  May 2010 Published  September 2010

In this paper we discuss the large time behavior of the solution to the Cauchy problem governed by a transport equation with Maxwell boundary conditions arising in growing cell population in $L^1$-spaces. Our result completes previous ones established in [3] in $L^p$-spaces with $1 < p < \infty$.
Citation: Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305
##### References:
 [1] B. Lods, On linear kinetic equations involving unbounded cross-sections,, Math. Meth. Appl. Sci., 27 (2004), 1049. doi: doi:10.1002/mma.485. Google Scholar [2] B. Lods, Semigroup generation properties of streaming operators with noncontractive boundary conditions,, Mathematical and Computer Modelling, 42 (2005), 1441. doi: doi:10.1016/j.mcm.2004.12.007. Google Scholar [3] B. Lods and M. Sbihi, Stability of the essential spectrum for $2D$-transport models with Maxwell boundary conditions,, Math. Meth. Appl. Sci., 29 (2006), 499. doi: doi:10.1002/mma.684. Google Scholar [4] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population,, J. Math. Biol., 1 (1974), 17. doi: doi:10.1007/BF02339486. Google Scholar [5] J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups,, Mh. Math., 90 (1980), 153. doi: doi:10.1007/BF01303264. Google Scholar [6] J. Voigt, Spectral properties of the neutron transport equation,, J. Math. Anal. Appl., 106 (1985), 140. doi: doi:10.1016/0022-247X(85)90137-4. Google Scholar [7] K. Latrach and A. Zeghal, Existence results for a boundary value problem arising in growing cell populations,, Math. Models Meth. Appl. Sci., 13 (2003), 1. doi: doi:10.1142/S0218202503002350. Google Scholar [8] K. Latrach and B. Lods, Regularity and time asymptotic behaviour of solutions to transport equations,, Transp. Theory Stat. Phys., 30 (2001), 617. doi: doi:10.1081/TT-100107419. Google Scholar [9] K. Latrach and H. Megdiche, Spectral properties and regularity of solutions to transport equations in slab geometry,, Math. Models Appl. Sci., 29 (2006), 2089. Google Scholar [10] K. Latrach, H. Megdiche and M. A. Taoudi, Compactness properties for perturbed semigroups in Banach spaces and application to a transport model,, J. Math. Anal. Appl., 359 (2009), 88. doi: doi:10.1016/j.jmaa.2009.05.027. Google Scholar [11] L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory,, J. Math. Anal. Appl., 129 (1988), 6. doi: doi:10.1016/0022-247X(88)90230-2. Google Scholar [12] L. W. Weis, The stability of positive semigroups on $L_p$ spaces,, Proc. AMS, 123 (1995), 3089. Google Scholar [13] M. Mokhtar-Kharroubi, On $L^1$-spectral theory of neutron transport,, J. Diff. Int. Equ., 18 (2005), 1221. Google Scholar [14] M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biol., 103 (1983), 181. doi: doi:10.1016/0022-5193(83)90024-3. Google Scholar [15] M. Sbihi, A resolvent approach to the stability of essential and critical spectra of perturbed $C_0$-semigroups on Hilbert spaces with applications to transport theory,, J. Evol. Equ., 7 (2007), 35. doi: doi:10.1007/s00028-006-0226-2. Google Scholar [16] P. Dodds and J. Fremlin, Compact operator in Banach lattices,, Isr. J. Math., 34 (1979), 287. doi: doi:10.1007/BF02760610. Google Scholar [17] H. Hille and R. E. Phillips, "Functional Analysis and Semigroups," Vol. 31,, Amer. Math. Soc. Colloq., (1957). Google Scholar [18] N. Dunford and J. T. Schwartz, "Linear Operators: Part I,", Intersciences, (1958). Google Scholar [19] R. Nagel (ed.), "One-Parameter Semigroups of Positive Operators,", Lect. Notes Math., (1184). Google Scholar [20] T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1966). Google Scholar [21] W. Greenberg, C. Van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkhäuser, (1987). Google Scholar

show all references

##### References:
 [1] B. Lods, On linear kinetic equations involving unbounded cross-sections,, Math. Meth. Appl. Sci., 27 (2004), 1049. doi: doi:10.1002/mma.485. Google Scholar [2] B. Lods, Semigroup generation properties of streaming operators with noncontractive boundary conditions,, Mathematical and Computer Modelling, 42 (2005), 1441. doi: doi:10.1016/j.mcm.2004.12.007. Google Scholar [3] B. Lods and M. Sbihi, Stability of the essential spectrum for $2D$-transport models with Maxwell boundary conditions,, Math. Meth. Appl. Sci., 29 (2006), 499. doi: doi:10.1002/mma.684. Google Scholar [4] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population,, J. Math. Biol., 1 (1974), 17. doi: doi:10.1007/BF02339486. Google Scholar [5] J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups,, Mh. Math., 90 (1980), 153. doi: doi:10.1007/BF01303264. Google Scholar [6] J. Voigt, Spectral properties of the neutron transport equation,, J. Math. Anal. Appl., 106 (1985), 140. doi: doi:10.1016/0022-247X(85)90137-4. Google Scholar [7] K. Latrach and A. Zeghal, Existence results for a boundary value problem arising in growing cell populations,, Math. Models Meth. Appl. Sci., 13 (2003), 1. doi: doi:10.1142/S0218202503002350. Google Scholar [8] K. Latrach and B. Lods, Regularity and time asymptotic behaviour of solutions to transport equations,, Transp. Theory Stat. Phys., 30 (2001), 617. doi: doi:10.1081/TT-100107419. Google Scholar [9] K. Latrach and H. Megdiche, Spectral properties and regularity of solutions to transport equations in slab geometry,, Math. Models Appl. Sci., 29 (2006), 2089. Google Scholar [10] K. Latrach, H. Megdiche and M. A. Taoudi, Compactness properties for perturbed semigroups in Banach spaces and application to a transport model,, J. Math. Anal. Appl., 359 (2009), 88. doi: doi:10.1016/j.jmaa.2009.05.027. Google Scholar [11] L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory,, J. Math. Anal. Appl., 129 (1988), 6. doi: doi:10.1016/0022-247X(88)90230-2. Google Scholar [12] L. W. Weis, The stability of positive semigroups on $L_p$ spaces,, Proc. AMS, 123 (1995), 3089. Google Scholar [13] M. Mokhtar-Kharroubi, On $L^1$-spectral theory of neutron transport,, J. Diff. Int. Equ., 18 (2005), 1221. Google Scholar [14] M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biol., 103 (1983), 181. doi: doi:10.1016/0022-5193(83)90024-3. Google Scholar [15] M. Sbihi, A resolvent approach to the stability of essential and critical spectra of perturbed $C_0$-semigroups on Hilbert spaces with applications to transport theory,, J. Evol. Equ., 7 (2007), 35. doi: doi:10.1007/s00028-006-0226-2. Google Scholar [16] P. Dodds and J. Fremlin, Compact operator in Banach lattices,, Isr. J. Math., 34 (1979), 287. doi: doi:10.1007/BF02760610. Google Scholar [17] H. Hille and R. E. Phillips, "Functional Analysis and Semigroups," Vol. 31,, Amer. Math. Soc. Colloq., (1957). Google Scholar [18] N. Dunford and J. T. Schwartz, "Linear Operators: Part I,", Intersciences, (1958). Google Scholar [19] R. Nagel (ed.), "One-Parameter Semigroups of Positive Operators,", Lect. Notes Math., (1184). Google Scholar [20] T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1966). Google Scholar [21] W. Greenberg, C. Van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkhäuser, (1987). Google Scholar
 [1] Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457 [2] Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123 [3] Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic & Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041 [4] Manuela Giampieri, Stefano Isola. A one-parameter family of analytic Markov maps with an intermittency transition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 115-136. doi: 10.3934/dcds.2005.12.115 [5] Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877 [6] Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016 [7] Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic & Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113 [8] Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139 [9] Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 [10] Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014 [11] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [12] Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019041 [13] Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 [14] Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133 [15] Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511 [16] Toshiyuki Ogawa, Takashi Okuda. Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273 [17] Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102 [18] Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475 [19] Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991 [20] Ziran Yin, Liwei Zhang. Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1387-1397. doi: 10.3934/jimo.2018100

2018 Impact Factor: 1.143