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Time asymptotic behaviour for unbounded linear operators arising in growing cell populations. (English) Zbl 1039.92033

Summary: This paper is concerned with the spectral analysis of a class of unbounded linear operators, originally proposed by M. Rotenberg [J. Theor. Biol. 96, 495–509 (1982); ibid. 103, 181–199 (1983)]. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a \(C^0\)-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible if the boundary operator is strictly positive. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.

MSC:

92D25 Population dynamics (general)
47A10 Spectrum, resolvent
47G20 Integro-differential operators
92C37 Cell biology
47N60 Applications of operator theory in chemistry and life sciences
47D60 \(C\)-semigroups, regularized semigroups
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[1] Anselone, A. P.; Palmer, J. M., Spectral analysis of collectively compact. Strongly convergent operator sequences, Pacific J. Math., 25, 423-431 (1968) · Zbl 0157.45203
[2] Arendt, W., Resolvent positive operators, Proc. London Math. Soc., 54, 321-349 (1987) · Zbl 0617.47029
[3] Batty, J. K.; Robinson, D. W., Positive one parameter semigroups on ordered Banach spaces, Acta Appl. Math., 1, 221-296 (1984) · Zbl 0554.47022
[4] Boulanouar, M., Un modèle de Rotenberg avec la loi à mémoire parfaite, C. R. Acad. Sci. Paris, 327, I, 965-968 (1998) · Zbl 0914.92014
[5] Boulanouar, M.; Leboucher, L., Une équation de transport dans la dynamique des populations cellulaires, C. R. Acad. Sci. Paris, 321, I, 305-308 (1995) · Zbl 0835.92025
[6] Cessenat, M., Théorèmes de trace \(L_1\) pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris, 299, I, 831-834 (1984) · Zbl 0568.46030
[7] Cessenat, M., Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris, 300, I, 89-92 (1985) · Zbl 0648.46028
[8] Clement, P., One Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam
[9] J. Diestel, J.J. Uhl Jr., Vector Measure, Amer. Math. Soc. Providence, RI, 1977.; J. Diestel, J.J. Uhl Jr., Vector Measure, Amer. Math. Soc. Providence, RI, 1977.
[10] Dodds, P.; Fremlin, J., Compact operator in Banach lattices, Israel J. Math., 34, 287-320 (1979) · Zbl 0438.47042
[11] N. Dunford, J.T. Schwartz, Linears Operators, Part 1, Interscience Publishers Inc., New York, 1958.; N. Dunford, J.T. Schwartz, Linears Operators, Part 1, Interscience Publishers Inc., New York, 1958.
[12] Greenberg, W.; Van Der Mee, C.; Protopopescu, V., Boundary Value Problems in Abstract Kinetic Theory (1987), Birkhauser: Birkhauser Basel · Zbl 0624.35003
[13] Kaper, H. G.; Lekkerkerker, C. G.; Hejtmanek, J., Spectral Methods in Linear Transport Theory (1982), Birkhauser: Birkhauser Basel · Zbl 0498.47001
[14] Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer Berlin · Zbl 0148.12601
[15] Lebowitz, J. L.; Rubinow, S. I., A theory for the age and generation time distribution of a microbial population, J. Math. Biol., 1, 17-36 (1974) · Zbl 0402.92023
[16] Lehner, J.; Wing, M., Solution of the linearized Boltzmann transport equation for the slab geometry, Duke Math. J., 23, 125-142 (1956) · Zbl 0070.34702
[17] Mika, J.; Stankiewicz, R.; Trombetti, T., Effective solution to the initial value problem in neutron thermalization theory, J. Math. Anal. Appl., 25, 149-161 (1969) · Zbl 0184.32002
[18] Mokhtar-Kharroubi, M., Time asymptotic behaviour and compactness in neutron transport theory, Europ. J. Mech. B Fluid, 11, 39-68 (1992)
[19] R. Nagel (Ed.), On One-parameter Semigroup of Positive Operators, Lecture Notes on Mathematics, Vol. 1184, Springer, Berlin, 1986.; R. Nagel (Ed.), On One-parameter Semigroup of Positive Operators, Lecture Notes on Mathematics, Vol. 1184, Springer, Berlin, 1986.
[20] Rotenberg, M., Theory of distributed quiescent state in the cell cycle, J. Theor. Biol., 96, 495-509 (1982)
[21] Rotenberg, M., Transport theory for growing cell populations, J. Theoret. Biol., 103, 181-199 (1983)
[22] Van der Mee, C.; Zweifel, P., A Fokker-Plank equation for growing cell populations, J. Math. Biol., 25, 61-72 (1987) · Zbl 0644.92019
[23] Vidav, I., Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator, J. Math. Anal. Appl., 22, 144-155 (1968) · Zbl 0155.19203
[24] A. Defici, A. Jeribi, K. Latrach, On a transport operator arising in growing cell populations I. Spectral analysis, Advances in Mathematics research pp. 149-175 (2002).; A. Defici, A. Jeribi, K. Latrach, On a transport operator arising in growing cell populations I. Spectral analysis, Advances in Mathematics research pp. 149-175 (2002).
[25] Latrach, K.; Mokhtar-Kharroubi, M., On an unbounded linear operator arising in the theory of growing cell population, J. Math. Anal. Appl., 211, 273-294 (1997) · Zbl 0896.92023
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