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

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