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Spectral analysis of semigroups and growth-fragmentation equations. (English) Zbl 1357.47044

Authors’ abstract: The aim of this paper is twofold.
(1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl’s theorems and the Krein-Rutman theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series in the spirit of those used in [C. Mouhot, Commun. Math. Phys. 261, No. 3, 629–672 (2006; Zbl 1113.82062); the first author and C. Mouhot, ibid. 288, No. 2, 431–502 (2009; Zbl 1178.82056); “Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation”, Preprint, arXiv:1412.7487; M. P. Gualdani et al., “Factorization of non-symmetric operators and exponential \(H\)-theorem”, Preprint, arXiv:1006.5523].
(2) On the other hand, we present the semigroup spectral analysis for three important classes of “growth-fragmentation” equations, namely, the cell division equation, the self-similar fragmentation equation and the McKendrick-Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated first eigenfunction or self-similar profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in [B. Perthame and L. Ryzhik, J. Differ. Equations 210, No. 1, 155–177 (2005; Zbl 1072.35195); P. Laurençot and B. Perthame, Commun. Math. Sci. 7, No. 2, 503–510 (2009; Zbl 1183.35038)] for the cell division model with (almost) constant total fragmentation rate and in [M. J. Cáceres et al., J. Math. Pures Appl. (9) 96, No. 4, 334–362 (2011; Zbl 1235.35034); Commun. Appl. Ind. Math. 1, No. 2, Article ID 590, 299–308 (2010; Zbl 1329.82064)] for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in [P. Michel et al., J. Math. Pures Appl. (9) 84, No. 9, 1235–1260 (2005; Zbl 1085.35042); M. Escobedo et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 1, 99–125 (2005; Zbl 1130.35025)] which have been established under similar assumptions to those made in the present work.

MSC:

47D06 One-parameter semigroups and linear evolution equations
35P15 Estimates of eigenvalues in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92D25 Population dynamics (general)
34G10 Linear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
45C05 Eigenvalue problems for integral equations
45K05 Integro-partial differential equations
92C37 Cell biology
82D60 Statistical mechanics of polymers
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References:

[1] Andreu, F.; Martínez, J.; Mazón, J. M., A spectral mapping theorem for perturbed strongly continuous semigroups, Math. Ann., 291, 3, 453-462 (1991) · Zbl 0787.47033
[2] Arendt, W., Kato’s equality and spectral decomposition for positive \(C_0\)-groups, Manuscr. Math., 40, 2-3, 277-298 (1982) · Zbl 0543.47032
[3] Arendt, W.; Grabosch, A.; Greiner, G.; Groh, U.; Lotz, H. P.; Moustakas, U.; Nagel, R.; Neubrander, F.; Schlotterbeck, U., One-Parameter Semigroups of Positive Operators, Lect. Notes Math., vol. 1184 (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0585.47030
[4] Arkeryd, L., Stability in \(L^1\) for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 103, 2, 151-167 (1988) · Zbl 0654.76074
[5] Arkeryd, L.; Esposito, R.; Pulvirenti, M., The Boltzmann equation for weakly inhomogeneous data, Commun. Math. Phys., 111, 3, 393-407 (1987) · Zbl 0663.76080
[6] Arnold, A.; Gamba, I. M.; Gualdani, M. P.; Mischler, S.; Mouhot, C.; Sparber, C., The Wigner-Fokker-Planck equation: stationary states and large time behavior, Math. Models Methods Appl. Sci., 22, 11, 1250034 (2012), 31 · Zbl 1253.82065
[7] Baccelli, F.; McDonald, D.; Reynier, J., A mean field model for multiple TCP connections through a buffer implementing red, Perform. Evol., 11, 77-97 (2002)
[8] Balagué, D.; Cañizo, J. A.; Gabriel, P., Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, 6, 2, 219-243 (2013) · Zbl 1270.35095
[9] Banasiak, J.; Arlotti, L., Perturbations of Positive Semigroups with Applications, Springer Monogr. Math. (2006), Springer-Verlag London Ltd.: Springer-Verlag London Ltd. London · Zbl 1097.47038
[10] Banasiak, J.; Pichór, K.; Rudnicki, R., Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119, 149-166 (2012) · Zbl 1301.47055
[11] Basse, B.; Baguley, B. C.; Marshall, E. S.; Joseph, W. R.; van Brunt, B.; Wake, G.; Wall, D. J.N., A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol., 47, 4, 295-312 (2003) · Zbl 1050.92028
[12] Bell, G.; Anderson, E., Cell growth and division: I. A mathematical model with applications to cell volume distribution in mammalian suspension cultures, Biophys. J., 8, 4, 329-351 (1967)
[13] Bertoin, J., The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5, 4, 395-416 (2003) · Zbl 1042.60042
[14] Beysens, D.; Campi, X.; Pefferkorn, E., Fragmentation Phenomena (1995), World Scientific: World Scientific Singapore
[15] Bobylëv, A. V., The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225, 6, 1041-1044 (1975)
[16] Brendle, S.; Nagel, R.; Poland, J., On the spectral mapping theorem for perturbed strongly continuous semigroups, Arch. Math. (Basel), 74, 5, 365-378 (2000) · Zbl 1040.47031
[17] Brezis, H., Analyse fonctionnelle. Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise (1983), Masson: Masson Paris · Zbl 0511.46001
[18] Cáceres, M. J.; Cañizo, J. A.; Mischler, S., Rate of convergence to self-similarity for the fragmentation equation in \(L^1\) spaces, Commun. Appl. Ind. Math., 1, 2, 299-308 (2010) · Zbl 1329.82064
[19] Cáceres, M. J.; Cañizo, J. A.; Mischler, S., Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96, 4, 334-362 (2011) · Zbl 1235.35034
[20] Calvez, V.; Doumic, M.; Gabriel, P., Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, J. Math. Pures Appl. (9), 98, 1, 1-27 (2012) · Zbl 1259.35151
[21] Calvez, V.; Lenuzza, N.; Doumic, M.; Deslys, J.-P.; Mouthon, F.; Perthame, B., Prion dynamics with size dependency-strain phenomena, J. Biol. Dyn., 4, 1, 28-42 (2010) · Zbl 1315.92039
[22] Calvez, V.; Lenuzza, N.; Oelz, D.; Deslys, J.-P.; Laurent, P.; Mouthon, F.; Perthame, B., Size distribution dependence of prion aggregates infectivity, Math. Biosci., 217, 1, 88-99 (2009) · Zbl 1161.92033
[23] Carleman, T., Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler, vol. 2 (1957), Almqvist & Wiksells Boktryckeri Ab: Almqvist & Wiksells Boktryckeri Ab Uppsala · Zbl 0077.23401
[26] Chipot, M., On the equations of age-dependent population dynamics, Arch. Ration. Mech. Anal., 82, 1, 13-25 (1983) · Zbl 0507.92015
[27] Dautray, R.; Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3: Spectral Theory and Applications (1990), Springer-Verlag: Springer-Verlag Berlin, (with the collaboration of Michel Artola and Michel Cessenat; translated from the French by John C. Amson) · Zbl 0683.35001
[28] Davies, E. B., One-Parameter Semigroups, London Math. Soc. Monogr. Ser., vol. 15 (1980), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: Academic Press Inc. [Harcourt Brace Jovanovich Publishers] London · Zbl 0457.47030
[29] Diekmann, O.; Heijmans, H. J.A. M.; Thieme, H. R., On the stability of the cell size distribution, J. Math. Biol., 19, 2, 227-248 (1984) · Zbl 0543.92021
[30] Dolbeault, J.; Mouhot, C.; Schmeiser, C., Hypocoercivity for kinetic equations conserving mass, Trans. Am. Math. Soc. (2015), in press, arXiv eprint 1005.1495 · Zbl 1342.82115
[31] Dolbeault, J.; Mouhot, C.; Schmeiser, C., Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347, 9-10, 511-516 (2009) · Zbl 1177.35054
[32] Doumic Jauffret, M.; Gabriel, P., Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20, 5, 757-783 (2010) · Zbl 1201.35086
[33] Dyson, F. J., The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev., 75, 486-502 (1949) · Zbl 0032.23702
[35] Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts Math., vol. 194 (2000), Springer-Verlag: Springer-Verlag New York, (with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt) · Zbl 0952.47036
[36] Escobedo, M.; Mischler, S.; Rodriguez Ricard, M., On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22, 1, 99-125 (2005) · Zbl 1130.35025
[37] Feller, W., On the integral equation of renewal theory, Ann. Math. Stat., 12, 243-267 (1941) · JFM 67.0370.02
[38] Feller, W., An Introduction to Probability Theory and its Applications, vol. II (1966), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0138.10207
[39] Filippov, A., On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6, 275-293 (1961) · Zbl 0242.60050
[40] Fredholm, I., Sur une classe d’équations fonctionnelles, Acta Math., 27, 1, 365-390 (1903) · JFM 34.0422.02
[41] Fredrickson, A. G.; Ramakrishna, D.; Tsuchiya, H. M., Statistics and dynamics of prokaryotic cell populations, Math. Biosci., 327-374 (1967) · Zbl 0152.18901
[42] Frobenius, G., Über Matrizen aus nicht negativen Elementen, 456-477 (1912), Sitzungsber. Königl. Preuss. Akad. Wiss. · JFM 43.0204.09
[43] Gallay, T.; Wayne, C. E., Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(R^2\), Arch. Ration. Mech. Anal., 163, 3, 209-258 (2002) · Zbl 1042.37058
[44] Gearhart, L., Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc., 236, 385-394 (1978) · Zbl 0326.47038
[45] Grabosch, A., Compactness properties and asymptotics of strongly coupled systems, J. Math. Anal. Appl., 187, 2, 411-437 (1994) · Zbl 0841.92013
[46] Grad, H., Principles of the kinetic theory of gases, (Flügge, S., Thermodynamik der Gase. Thermodynamik der Gase, Handbuch der Physik, Bd. 12 (1958), Springer-Verlag: Springer-Verlag Berlin), 205-294
[47] Grad, H., Asymptotic theory of the Boltzmann equation. II, (Rarefied Gas Dynamics, vol. I (1963), Academic Press: Academic Press New York), 26-59, (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962)
[48] Greer, M. L.; Pujo-Menjouet, L.; Webb, G. F., A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol., 242, 3, 598-606 (2006) · Zbl 1447.92089
[49] Greiner, G., Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen, Math. Z., 177, 3, 401-423 (1981) · Zbl 0461.47016
[50] Greiner, G.; Voigt, J.; Wolff, M., On the spectral bound of the generator of semigroups of positive operators, J. Oper. Theory, 5, 2, 245-256 (1981) · Zbl 0469.47032
[52] Gurtin, M. E.; MacCamy, R. C., Non-linear age-dependent population dynamics, Arch. Ration. Mech. Anal., 54, 281-300 (1974) · Zbl 0286.92005
[53] Gurtin, M. E.; MacCamy, R. C., Some simple models for nonlinear age-dependent population dynamics, Math. Biosci., 43, 3-4, 199-211 (1979) · Zbl 0397.92025
[54] Gyllenberg, M.; Webb, G. F., A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28, 6, 671-694 (1990) · Zbl 0744.92026
[55] Heijmans, H. J.A. M., On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Biosci., 72, 1, 19-50 (1984) · Zbl 0568.92015
[56] Helffer, B.; Nier, F., Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lect. Notes Math., vol. 1862 (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1072.35006
[57] Hérau, F., Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46, 3-4, 349-359 (2006) · Zbl 1096.35019
[58] Hérau, F., Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244, 1, 95-118 (2007) · Zbl 1120.35016
[59] Hérau, F.; Nier, F., Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171, 2, 151-218 (2004) · Zbl 1139.82323
[60] Hilbert, D., Begründung der kinetischen Gastheorie, Math. Ann., 72, 4, 562-577 (1912) · JFM 43.1055.03
[61] Hilbert, D., Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1953), Chelsea Publishing Company: Chelsea Publishing Company New York, NY · Zbl 0050.10201
[62] Hille, E., Representation of one-parameter semigroups of linear transformations, Proc. Natl. Acad. Sci. USA, 28, 175-178 (1942)
[63] Hille, E., Functional Analysis and Semi-Groups, Colloq. Publ. - Am. Math. Soc., vol. 31 (1948), American Mathematical Society: American Mathematical Society New York · Zbl 0033.06501
[64] Hille, E.; Phillips, R. S., Functional Analysis and Semi-Groups, Colloq. Publ. - Am. Math. Soc., vol. 31 (1957), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0078.10004
[65] Iannelli, M., Mathematical Theory of Age-Structured Population Dynamics (1994), Giardini Editori and Stampore: Giardini Editori and Stampore Pisa, Italy
[66] Jörgens, K., An asymptotic expansion in the theory of neutron transport, Commun. Pure Appl. Math., 11, 219-242 (1958) · Zbl 0081.44105
[67] Kato, T., Perturbation Theory for Linear Operators, Class. Math. (1995), Springer-Verlag: Springer-Verlag Berlin, Reprint of the 1980 edition · Zbl 0836.47009
[68] Koch, A.; Schaechter, M., A model for statistics of the cell division process, J. Gen. Microbiol., 29, 435-454 (1962)
[69] Komatsu, H., Fractional powers of operators, Pac. J. Math., 19, 285-346 (1966) · Zbl 0154.16104
[70] Komatsu, H., Fractional powers of operators. II. Interpolation spaces, Pac. J. Math., 21, 89-111 (1967) · Zbl 0168.10702
[71] Kreĭn, M. G.; Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Usp. Mat. Nauk (N.S.), 3, 1(23), 3-95 (1948) · Zbl 0030.12902
[72] Latrach, K., Compactness properties for perturbed semigroups and application to transport equation, J. Aust. Math. Soc. A, 69, 1, 25-40 (2000) · Zbl 0970.47007
[73] Laurençot, P.; Perthame, B., Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7, 2, 503-510 (2009) · Zbl 1183.35038
[74] Leslie, P. H., On the use of matrices in certain population mathematics, Biometrika, 33, 183-212 (1945) · Zbl 0060.31803
[75] Liapunov, A. M., Stability of Motion, Math. Sci. Eng., vol. 30 (1966), Academic Press: Academic Press New York, (with a contribution by V.A. Pliss and an introduction by V.P. Basov; translated from the Russian by Flavian Abramovici and Michael Shimshoni) · Zbl 0161.06303
[76] Lions, J.-L.; Peetre, J., Sur une classe d’espaces d’interpolation, Publ. Math. IHÉS, 19, 5-68 (1964) · Zbl 0148.11403
[77] Lotka, A. J.; Sharpe, F., A problem in age distribution, Philos. Mag., 21, 435-438 (1911) · JFM 42.1030.02
[78] Lumer, G.; Phillips, R. S., Dissipative operators in a Banach space, Pac. J. Math., 11, 679-698 (1961) · Zbl 0101.09503
[79] McKendrick, A., Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44, 98-130 (1926) · JFM 52.0542.04
[80] Melzak, Z. A., A scalar transport equation, Trans. Am. Math. Soc., 85, 547-560 (1957) · Zbl 0077.30505
[81] (Metz, J. A.J.; Diekmann, O., The Dynamics of Physiologically Structured Populations. The Dynamics of Physiologically Structured Populations, Lect. Notes Biomath., vol. 68 (1986), Springer-Verlag: Springer-Verlag Berlin), (Papers from the colloquium held in Amsterdam, 1983) · Zbl 0614.92014
[82] Michel, P., Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16, 7, suppl., 1125-1153 (2006) · Zbl 1094.92023
[83] Michel, P.; Mischler, S.; Perthame, B., General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338, 9, 697-702 (2004) · Zbl 1049.35070
[84] Michel, P.; Mischler, S.; Perthame, B., General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl. (9), 84, 9, 1235-1260 (2005) · Zbl 1085.35042
[86] Mischler, S.; Mouhot, C., Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation · Zbl 1338.35430
[87] Mischler, S.; Mouhot, C., Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Commun. Math. Phys., 288, 2, 431-502 (2009) · Zbl 1178.82056
[88] Mischler, S.; Mouhot, C., Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media, Discrete Contin. Dyn. Syst., 24, 1, 159-185 (2009) · Zbl 1160.76042
[89] Mischler, S.; Perthame, B.; Ryzhik, L., Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., 12, 12, 1751-1772 (2002) · Zbl 1020.92025
[90] Miyadera, I., On perturbation theory for semi-groups of operators, Tôhoku Math. J. (2), 18, 299-310 (1966) · Zbl 0193.10902
[91] Mokhtar-Kharroubi, M., Compactness properties for positive semigroups on Banach lattices and applications, Houst. J. Math., 17, 1, 25-38 (1991) · Zbl 0744.47035
[92] Mokhtar-Kharroubi, M., Time asymptotic behavior and compactness in transport theory, Eur. J. Mech. B, Fluids, 11, 1, 39-68 (1992)
[93] Mokhtar-Kharroubi, M., Mathematical Topics in Neutron Transport Theory, Ser. Adv. Math. Appl. Sci., vol. 46 (1997), World Scientific Publishing Co. Inc.: World Scientific Publishing Co. Inc. River Edge, NJ, (New aspects, with a chapter by M. Choulli and P. Stefanov) · Zbl 0997.82047
[94] Mokhtar-Kharroubi, M., Spectral properties of a class of positive semigroups on Banach lattices and streaming operators, Positivity, 10, 2, 231-249 (2006) · Zbl 1107.47030
[95] Mokhtar-Kharroubi, M., On \(L^1\) exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 266, 11, 6418-6455 (2014) · Zbl 1304.47055
[96] Mouhot, C., Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Commun. Math. Phys., 261, 3, 629-672 (2006) · Zbl 1113.82062
[97] Mouhot, C.; Neumann, L., Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19, 4, 969-998 (2006) · Zbl 1169.82306
[98] Nagel, R.; Uhlig, H., An abstract Kato inequality for generators of positive operators semigroups on Banach lattices, J. Oper. Theory, 6, 1, 113-123 (1981) · Zbl 0486.47025
[99] Painter, P.; Marr, A., Mathematics of microbial populations, Annu. Rev. Microbiol., 22, 519-548 (1968)
[100] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44 (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[101] Perron, O., Zur Theorie der Matrices, Math. Ann., 64, 2, 248-263 (1907) · JFM 38.0202.01
[102] Perthame, B., Transport Equations in Biology, Front. Math. (2007), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1185.92006
[103] Perthame, B.; Ryzhik, L., Exponential decay for the fragmentation or cell-division equation, J. Differ. Equ., 210, 1, 155-177 (2005) · Zbl 1072.35195
[104] Phillips, R. S., Spectral theory for semi-groups of linear operators, Trans. Am. Math. Soc., 71, 393-415 (1951) · Zbl 0045.21502
[105] Phillips, R. S., Perturbation theory for semi-groups of linear operators, Trans. Am. Math. Soc., 74, 199-221 (1953) · Zbl 0053.08704
[106] Phillips, R. S., Semi-groups of positive contraction operators, Czechoslov. Math. J., 12, 87, 294-313 (1962) · Zbl 0113.09901
[107] Prüss, J., On the spectrum of \(C_0\)-semigroups, Trans. Am. Math. Soc., 284, 2, 847-857 (1984) · Zbl 0572.47030
[108] Prüss, J.; Pujo-Menjouet, L.; Webb, G. F.; Zacher, R., Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst., Ser. B, 6, 1, 225-235 (2006) · Zbl 1088.92043
[109] Ribarič, M.; Vidav, I., Analytic properties of the inverse \(A(z)^{- 1}\) of an analytic linear operator valued function \(A(z)\), Arch. Ration. Mech. Anal., 32, 298-310 (1969) · Zbl 0174.18002
[110] Rudnicki, R.; Pichór, K., Markov semigroups and stability of the cell maturity distribution, J. Biol. Syst., 8, 1, 69-94 (2000)
[111] Sinko, J.; Streifer, W., A model for populations peproducting by fission, Ecology, 52, 2, 330-335 (1971)
[113] Ukai, S., On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Jpn. Acad., 50, 179-184 (1974) · Zbl 0312.35061
[114] Vidav, I., Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30, 264-279 (1970) · Zbl 0195.13704
[115] Villani, C., Hypocoercive diffusion operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 10, 2, 257-275 (2007) · Zbl 1178.35306
[116] Villani, C., Hypocoercivity, Mem. Am. Math. Soc., 202, 950, iv+141 (2009) · Zbl 1197.35004
[117] Voigt, J., On the perturbation theory for strongly continuous semigroups, Math. Ann., 229, 2, 163-171 (1977) · Zbl 0338.47018
[118] Voigt, J., A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatshefte Math., 90, 2, 153-161 (1980) · Zbl 0433.47022
[119] von Foerster, H. Some remarks on changing population, (Stohlman, F., The Kinetics of Cell Proliferation (1959), Grune and Stratton: Grune and Stratton New York), 382-407
[120] Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics, Monogr. Textb. Pure Appl. Math., vol. 89 (1985), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0555.92014
[121] Wennberg, B., Stability and exponential convergence for the Boltzmann equation, Arch. Ration. Mech. Anal., 130, 2, 103-144 (1995) · Zbl 0828.76076
[122] Weyl, H., Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann., 68, 2, 220-269 (1910) · JFM 41.0343.01
[123] Yosida, K., On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Jpn., 1, 15-21 (1948) · Zbl 0037.35302
[124] Ziff, R. M.; McGrady, E. D., The kinetics of cluster fragmentation and depolymerisation, J. Phys. A, 18, 15, 3027-3037 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.