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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

L. W. Weis, The stability of positive semigroups on $L_p$ spaces,, Proc. AMS, 123 (1995), 3089.

[13]

M. Mokhtar-Kharroubi, On $L^1$-spectral theory of neutron transport,, J. Diff. Int. Equ., 18 (2005), 1221.

[14]

M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biol., 103 (1983), 181. doi: doi:10.1016/0022-5193(83)90024-3.

[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.

[16]

P. Dodds and J. Fremlin, Compact operator in Banach lattices,, Isr. J. Math., 34 (1979), 287. doi: doi:10.1007/BF02760610.

[17]

H. Hille and R. E. Phillips, "Functional Analysis and Semigroups," Vol. 31,, Amer. Math. Soc. Colloq., (1957).

[18]

N. Dunford and J. T. Schwartz, "Linear Operators: Part I,", Intersciences, (1958).

[19]

R. Nagel (ed.), "One-Parameter Semigroups of Positive Operators,", Lect. Notes Math., (1184).

[20]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1966).

[21]

W. Greenberg, C. Van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkhäuser, (1987).

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

L. W. Weis, The stability of positive semigroups on $L_p$ spaces,, Proc. AMS, 123 (1995), 3089.

[13]

M. Mokhtar-Kharroubi, On $L^1$-spectral theory of neutron transport,, J. Diff. Int. Equ., 18 (2005), 1221.

[14]

M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biol., 103 (1983), 181. doi: doi:10.1016/0022-5193(83)90024-3.

[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.

[16]

P. Dodds and J. Fremlin, Compact operator in Banach lattices,, Isr. J. Math., 34 (1979), 287. doi: doi:10.1007/BF02760610.

[17]

H. Hille and R. E. Phillips, "Functional Analysis and Semigroups," Vol. 31,, Amer. Math. Soc. Colloq., (1957).

[18]

N. Dunford and J. T. Schwartz, "Linear Operators: Part I,", Intersciences, (1958).

[19]

R. Nagel (ed.), "One-Parameter Semigroups of Positive Operators,", Lect. Notes Math., (1184).

[20]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1966).

[21]

W. Greenberg, C. Van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkhäuser, (1987).

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