Spectral analysis of semigroups and growth-fragmentation equations
Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 3, pp. 849-898.

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 [96,87,51,86].

(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 [103,73] for the cell division model with (almost) constant total fragmentation rate and in [19,18] 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 [84,36] which have been established under similar assumptions to those made in the present work.

DOI: 10.1016/j.anihpc.2015.01.007
Classification: 47D06, 35P15, 35B40, 92D25, 34G10, 34K30, 35P05, 45C05, 47A10, 45C05, 45K05, 35410, 92C37, 82D60
Keywords: Spectral mapping theorem, Weyl's Theorem, Krein–Rutman theorem, Growth-fragmentation equations, Self-similarity, Exponential rate of convergence
@article{AIHPC_2016__33_3_849_0,
     author = {Mischler, S. and Scher, J.},
     title = {Spectral analysis of semigroups and growth-fragmentation equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {849--898},
     publisher = {Elsevier},
     volume = {33},
     number = {3},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.01.007},
     zbl = {1357.47044},
     mrnumber = {3489637},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.007/}
}
TY  - JOUR
AU  - Mischler, S.
AU  - Scher, J.
TI  - Spectral analysis of semigroups and growth-fragmentation equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 849
EP  - 898
VL  - 33
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.007/
DO  - 10.1016/j.anihpc.2015.01.007
LA  - en
ID  - AIHPC_2016__33_3_849_0
ER  - 
%0 Journal Article
%A Mischler, S.
%A Scher, J.
%T Spectral analysis of semigroups and growth-fragmentation equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 849-898
%V 33
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.007/
%R 10.1016/j.anihpc.2015.01.007
%G en
%F AIHPC_2016__33_3_849_0
Mischler, S.; Scher, J. Spectral analysis of semigroups and growth-fragmentation equations. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 3, pp. 849-898. doi : 10.1016/j.anihpc.2015.01.007. http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.007/

[1] Andreu, F.; Martínez, J.; Mazón, J.M. A spectral mapping theorem for perturbed strongly continuous semigroups, Math. Ann., Volume 291 (1991) no. 3, pp. 453–462 | MR | Zbl

[2] Arendt, W. Kato's equality and spectral decomposition for positive C0-groups, Manuscr. Math., Volume 40 (1982) no. 2–3, pp. 277–298 | MR | Zbl

[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, Springer-Verlag, Berlin, 1986 | DOI | MR

[4] Arkeryd, L. Stability in L1 for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., Volume 103 (1988) no. 2, pp. 151–167 | DOI | MR | Zbl

[5] Arkeryd, L.; Esposito, R.; Pulvirenti, M. The Boltzmann equation for weakly inhomogeneous data, Commun. Math. Phys., Volume 111 (1987) no. 3, pp. 393–407 | DOI | MR | Zbl

[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., Volume 22 (2012) no. 11, pp. 1250034 (31) | DOI | MR | Zbl

[7] Baccelli, F.; McDonald, D.; Reynier, J. A mean field model for multiple TCP connections through a buffer implementing red, Perform. Evol., Volume 11 (2002), pp. 77–97

[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, Volume 6 (2013) no. 2, pp. 219–243 | DOI | MR | Zbl

[9] Banasiak, J.; Arlotti, L. Perturbations of Positive Semigroups with Applications, Springer Monogr. Math., Springer-Verlag London Ltd., London, 2006 | MR | Zbl

[10] Banasiak, J.; Pichór, K.; Rudnicki, R. Asynchronous exponential growth of a general structured population model, Acta Appl. Math., Volume 119 (2012), pp. 149–166 | DOI | MR | Zbl

[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., Volume 47 (2003) no. 4, pp. 295–312 | DOI | MR | Zbl

[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., Volume 8 (1967) no. 4, pp. 329–351 | DOI

[13] Bertoin, J. The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., Volume 5 (2003) no. 4, pp. 395–416 | DOI | MR | Zbl

[14] Beysens, D.; Campi, X.; Pefferkorn, E. Fragmentation Phenomena, World Scientific, Singapore, 1995

[15] Bobylëv, A.V. The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, Volume 225 (1975) no. 6, pp. 1041–1044 | MR | Zbl

[16] Brendle, S.; Nagel, R.; Poland, J. On the spectral mapping theorem for perturbed strongly continuous semigroups, Arch. Math. (Basel), Volume 74 (2000) no. 5, pp. 365–378 | DOI | MR | Zbl

[17] Brezis, H. Analyse fonctionnelle. Théorie et applications, Theory and applications, Collection of Applied Mathematics for the Master's Degree, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983 | MR | Zbl

[18] Cáceres, M.J.; Cañizo, J.A.; Mischler, S. Rate of convergence to self-similarity for the fragmentation equation in L1 spaces, Commun. Appl. Ind. Math., Volume 1 (2010) no. 2, pp. 299–308 | MR

[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), Volume 96 (2011) no. 4, pp. 334–362 | DOI | MR | Zbl

[20] Calvez, V.; Doumic, M.; Gabriel, P. Self-similarity in a general aggregation–fragmentation problem. Application to fitness analysis, J. Math. Pures Appl. (9), Volume 98 (2012) no. 1, pp. 1–27 | DOI | MR | Zbl

[21] Calvez, V.; Lenuzza, N.; Doumic, M.; Deslys, J.-P.; Mouthon, F.; Perthame, B. Prion dynamics with size dependency–strain phenomena, J. Biol. Dyn., Volume 4 (2010) no. 1, pp. 28–42 | DOI | MR | Zbl

[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., Volume 217 (2009) no. 1, pp. 88–99 | DOI | MR | Zbl

[23] Carleman, T. Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler, vol. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957 | MR | Zbl

[24] K. Carrapatoso, Exponential convergence to equilibrium for the homogeneous Landau equation, preprint.

[25] K. Carrapatoso, G. Egaña, S. Mischler, Uniqueness and long time asymptotic for the Keller–Segel equation – part II, the parabolic–parabolic case, in progress.

[26] Chipot, M. On the equations of age-dependent population dynamics, Arch. Ration. Mech. Anal., Volume 82 (1983) no. 1, pp. 13–25 | DOI | MR | Zbl

[27] Dautray, R.; Lions, J.-L. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3: Spectral Theory and Applications, Springer-Verlag, Berlin, 1990 (with the collaboration of Michel Artola and Michel Cessenat; translated from the French by John C. Amson) | MR | Zbl

[28] Davies, E.B. One-Parameter Semigroups, London Math. Soc. Monogr. Ser., vol. 15, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1980 | MR | Zbl

[29] Diekmann, O.; Heijmans, H.J.A.M.; Thieme, H.R. On the stability of the cell size distribution, J. Math. Biol., Volume 19 (1984) no. 2, pp. 227–248 | DOI | MR | Zbl

[30] Dolbeault, J.; Mouhot, C.; Schmeiser, C. Hypocoercivity for kinetic equations conserving mass, Trans. Am. Math. Soc. (2015) (in press, arXiv eprint 1005.1495) | DOI | MR | Zbl

[31] Dolbeault, J.; Mouhot, C.; Schmeiser, C. Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 9–10, pp. 511–516 | MR | Zbl

[32] Doumic Jauffret, M.; Gabriel, P. Eigenelements of a general aggregation–fragmentation model, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 5, pp. 757–783 | DOI | MR | Zbl

[33] Dyson, F.J. The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev., Volume 75 (1949), pp. 486–502 | DOI | MR | Zbl

[34] G. Egaña, S. Mischler, Uniqueness and long time asymptotic for the Keller–Segel equation – part I. The parabolic–elliptic case, preprint.

[35] Engel, K.-J.; Nagel, R. One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts Math., vol. 194, Springer-Verlag, New York, 2000 (with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt) | MR | Zbl

[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, Volume 22 (2005) no. 1, pp. 99–125 | DOI | Numdam | MR | Zbl

[37] Feller, W. On the integral equation of renewal theory, Ann. Math. Stat., Volume 12 (1941), pp. 243–267 | DOI | JFM | MR | Zbl

[38] Feller, W. An Introduction to Probability Theory and its Applications, vol. II, John Wiley & Sons Inc., New York, 1966 | MR | Zbl

[39] Filippov, A. On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., Volume 6 (1961), pp. 275–293 | DOI | Zbl

[40] Fredholm, I. Sur une classe d'équations fonctionnelles, Acta Math., Volume 27 (1903) no. 1, pp. 365–390 | JFM | MR

[41] Fredrickson, A.G.; Ramakrishna, D.; Tsuchiya, H.M. Statistics and dynamics of prokaryotic cell populations, Math. Biosci. (1967), pp. 327–374 | DOI | Zbl

[42] Frobenius, G., Sitzungsber. Königl. Preuss. Akad. Wiss. (1912), pp. 456–477 | JFM

[43] Gallay, T.; Wayne, C.E. Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on R2 , Arch. Ration. Mech. Anal., Volume 163 (2002) no. 3, pp. 209–258 | DOI | MR | Zbl

[44] Gearhart, L. Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385–394 | DOI | MR | Zbl

[45] Grabosch, A. Compactness properties and asymptotics of strongly coupled systems, J. Math. Anal. Appl., Volume 187 (1994) no. 2, pp. 411–437 | DOI | MR | Zbl

[46] Grad, H.; Flügge, S. Principles of the kinetic theory of gases, Thermodynamik der Gase, Handbuch der Physik, Bd. 12, Springer-Verlag, Berlin, 1958, pp. 205–294 | MR

[47] Grad, H. Asymptotic theory of the Boltzmann equation. II, Rarefied Gas Dynamics, vol. I, Academic Press, New York, 1963, pp. 26–59 (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962) | MR

[48] Greer, M.L.; Pujo-Menjouet, L.; Webb, G.F. A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol., Volume 242 (2006) no. 3, pp. 598–606 | DOI | MR | Zbl

[49] Greiner, G. Zur Perron–Frobenius-Theorie stark stetiger Halbgruppen, Math. Z., Volume 177 (1981) no. 3, pp. 401–423 | DOI | MR | Zbl

[50] Greiner, G.; Voigt, J.; Wolff, M. On the spectral bound of the generator of semigroups of positive operators, J. Oper. Theory, Volume 5 (1981) no. 2, pp. 245–256 | MR | Zbl

[51] M.P. Gualdani, S. Mischler, C. Mouhot, Factorization of non-symmetric operators and exponential H-theorem, hal-00495786.

[52] Gurtin, M.E.; MacCamy, R.C. Non-linear age-dependent population dynamics, Arch. Ration. Mech. Anal., Volume 54 (1974), pp. 281–300 | DOI | MR | Zbl

[53] Gurtin, M.E.; MacCamy, R.C. Some simple models for nonlinear age-dependent population dynamics, Math. Biosci., Volume 43 (1979) no. 3–4, pp. 199–211 | MR | Zbl

[54] Gyllenberg, M.; Webb, G.F. A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., Volume 28 (1990) no. 6, pp. 671–694 | DOI | MR | Zbl

[55] Heijmans, H.J.A.M. On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Biosci., Volume 72 (1984) no. 1, pp. 19–50 | MR | Zbl

[56] Helffer, B.; Nier, F. Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Lect. Notes Math., vol. 1862, Springer-Verlag, Berlin, 2005 | MR | Zbl

[57] Hérau, F. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., Volume 46 (2006) no. 3–4, pp. 349–359 | MR | Zbl

[58] Hérau, F. Short and long time behavior of the Fokker–Planck equation in a confining potential and applications, J. Funct. Anal., Volume 244 (2007) no. 1, pp. 95–118 | DOI | MR | Zbl

[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., Volume 171 (2004) no. 2, pp. 151–218 | DOI | MR | Zbl

[60] Hilbert, D. Begründung der kinetischen Gastheorie, Math. Ann., Volume 72 (1912) no. 4, pp. 562–577 | DOI | JFM | MR

[61] Hilbert, D. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Publishing Company, New York, NY, 1953 | MR | Zbl

[62] Hille, E. Representation of one-parameter semigroups of linear transformations, Proc. Natl. Acad. Sci. USA, Volume 28 (1942), pp. 175–178 | DOI | MR

[63] Hille, E. Functional Analysis and Semi-Groups, Colloq. Publ. – Am. Math. Soc., vol. 31, American Mathematical Society, New York, 1948 | MR | Zbl

[64] Hille, E.; Phillips, R.S. Functional Analysis and Semi-Groups, Colloq. Publ. – Am. Math. Soc., vol. 31, American Mathematical Society, Providence, RI, 1957 | MR | Zbl

[65] Iannelli, M. Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori and Stampore, Pisa, Italy, 1994

[66] Jörgens, K. An asymptotic expansion in the theory of neutron transport, Commun. Pure Appl. Math., Volume 11 (1958), pp. 219–242 | DOI | MR | Zbl

[67] Kato, T. Perturbation Theory for Linear Operators, Class. Math., Springer-Verlag, Berlin, 1995 (Reprint of the 1980 edition) | MR | Zbl

[68] Koch, A.; Schaechter, M. A model for statistics of the cell division process, J. Gen. Microbiol., Volume 29 (1962), pp. 435–454 | DOI

[69] Komatsu, H. Fractional powers of operators, Pac. J. Math., Volume 19 (1966), pp. 285–346 | DOI | MR | Zbl

[70] Komatsu, H. Fractional powers of operators. II. Interpolation spaces, Pac. J. Math., Volume 21 (1967), pp. 89–111 | DOI | MR | Zbl

[71] Kreĭn, M.G.; Rutman, M.A. Linear operators leaving invariant a cone in a Banach space, Usp. Mat. Nauk (N.S.), Volume 3 (1948) no. 1(23), pp. 3–95 | MR | Zbl

[72] Latrach, K. Compactness properties for perturbed semigroups and application to transport equation, J. Aust. Math. Soc. A, Volume 69 (2000) no. 1, pp. 25–40 | DOI | MR | Zbl

[73] Laurençot, P.; Perthame, B. Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., Volume 7 (2009) no. 2, pp. 503–510 | DOI | MR | Zbl

[74] Leslie, P.H. On the use of matrices in certain population mathematics, Biometrika, Volume 33 (1945), pp. 183–212 | DOI | MR | Zbl

[75] Liapunov, A.M. Stability of Motion, Math. Sci. Eng., vol. 30, Academic Press, New York, 1966 (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

[76] Lions, J.-L.; Peetre, J. Sur une classe d'espaces d'interpolation, Publ. Math. IHÉS, Volume 19 (1964), pp. 5–68 | Numdam | Zbl

[77] Lotka, A.J.; Sharpe, F. A problem in age distribution, Philos. Mag., Volume 21 (1911), pp. 435–438 | JFM

[78] Lumer, G.; Phillips, R.S. Dissipative operators in a Banach space, Pac. J. Math., Volume 11 (1961), pp. 679–698 | DOI | Zbl

[79] McKendrick, A. Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., Volume 44 (1926), pp. 98–130 | JFM

[80] Melzak, Z.A. A scalar transport equation, Trans. Am. Math. Soc., Volume 85 (1957), pp. 547–560 | DOI | Zbl

[81] Metz, J.A.J.; Diekmann, O. The Dynamics of Physiologically Structured Populations, Lect. Notes Biomath., vol. 68, Springer-Verlag, Berlin, 1986 (Papers from the colloquium held in Amsterdam, 1983) | DOI | Zbl

[82] Michel, P. Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., Volume 16 (2006) no. 7, suppl., pp. 1125–1153 | Zbl

[83] Michel, P.; Mischler, S.; Perthame, B. General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 9, pp. 697–702 | DOI | Zbl

[84] Michel, P.; Mischler, S.; Perthame, B. General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl. (9), Volume 84 (2005) no. 9, pp. 1235–1260 | DOI | Zbl

[85] S. Mischler, Semigroup in Banach space, work in progress.

[86] Mischler, S.; Mouhot, C. Exponential stability of slowly decaying solutions to the kinetic Fokker–Planck equation | arXiv | DOI | Zbl

[87] Mischler, S.; Mouhot, C. Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Commun. Math. Phys., Volume 288 (2009) no. 2, pp. 431–502 | DOI | Zbl

[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., Volume 24 (2009) no. 1, pp. 159–185 | DOI | Zbl

[89] Mischler, S.; Perthame, B.; Ryzhik, L. Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., Volume 12 (2002) no. 12, pp. 1751–1772 | DOI | Zbl

[90] Miyadera, I. On perturbation theory for semi-groups of operators, Tôhoku Math. J. (2), Volume 18 (1966), pp. 299–310 | DOI | Zbl

[91] Mokhtar-Kharroubi, M. Compactness properties for positive semigroups on Banach lattices and applications, Houst. J. Math., Volume 17 (1991) no. 1, pp. 25–38 | Zbl

[92] Mokhtar-Kharroubi, M. Time asymptotic behavior and compactness in transport theory, Eur. J. Mech. B, Fluids, Volume 11 (1992) no. 1, pp. 39–68

[93] Mokhtar-Kharroubi, M. Mathematical Topics in Neutron Transport Theory, Ser. Adv. Math. Appl. Sci., vol. 46, World Scientific Publishing Co. Inc., River Edge, NJ, 1997 (New aspects, with a chapter by M. Choulli and P. Stefanov) | DOI | Zbl

[94] Mokhtar-Kharroubi, M. Spectral properties of a class of positive semigroups on Banach lattices and streaming operators, Positivity, Volume 10 (2006) no. 2, pp. 231–249 | DOI | Zbl

[95] Mokhtar-Kharroubi, M. On L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., Volume 266 (2014) no. 11, pp. 6418–6455 | DOI | Zbl

[96] Mouhot, C. Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Commun. Math. Phys., Volume 261 (2006) no. 3, pp. 629–672 | DOI | Zbl

[97] Mouhot, C.; Neumann, L. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, Volume 19 (2006) no. 4, pp. 969–998 | DOI | Zbl

[98] Nagel, R.; Uhlig, H. An abstract Kato inequality for generators of positive operators semigroups on Banach lattices, J. Oper. Theory, Volume 6 (1981) no. 1, pp. 113–123 | Zbl

[99] Painter, P.; Marr, A. Mathematics of microbial populations, Annu. Rev. Microbiol., Volume 22 (1968), pp. 519–548 | DOI

[100] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983 | Zbl

[101] Perron, O. Zur Theorie der Matrices, Math. Ann., Volume 64 (1907) no. 2, pp. 248–263 | DOI | JFM

[102] Perthame, B. Transport Equations in Biology, Front. Math., Birkhäuser Verlag, Basel, 2007 | DOI | Zbl

[103] Perthame, B.; Ryzhik, L. Exponential decay for the fragmentation or cell-division equation, J. Differ. Equ., Volume 210 (2005) no. 1, pp. 155–177 | DOI | Zbl

[104] Phillips, R.S. Spectral theory for semi-groups of linear operators, Trans. Am. Math. Soc., Volume 71 (1951), pp. 393–415 | DOI | Zbl

[105] Phillips, R.S. Perturbation theory for semi-groups of linear operators, Trans. Am. Math. Soc., Volume 74 (1953), pp. 199–221 | DOI | Zbl

[106] Phillips, R.S. Semi-groups of positive contraction operators, Czechoslov. Math. J., Volume 12 (1962) no. 87, pp. 294–313 | Zbl

[107] Prüss, J. On the spectrum of C0-semigroups, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 847–857 | Zbl

[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, Volume 6 (2006) no. 1, pp. 225–235 | Zbl

[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., Volume 32 (1969), pp. 298–310 | DOI | Zbl

[110] Rudnicki, R.; Pichór, K. Markov semigroups and stability of the cell maturity distribution, J. Biol. Syst., Volume 8 (2000) no. 1, pp. 69–94 | DOI

[111] Sinko, J.; Streifer, W. A model for populations peproducting by fission, Ecology, Volume 52 (1971) no. 2, pp. 330–335 | DOI

[112] I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogenous setting, preprint.

[113] Ukai, S. On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Jpn. Acad., Volume 50 (1974), pp. 179–184 | Zbl

[114] Vidav, I. Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., Volume 30 (1970), pp. 264–279 | DOI | Zbl

[115] Villani, C. Hypocoercive diffusion operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., Volume 10 (2007) no. 2, pp. 257–275 | Zbl

[116] Villani, C. Hypocoercivity, Mem. Am. Math. Soc., Volume 202 (2009) no. 950, pp. iv+141 | Zbl

[117] Voigt, J. On the perturbation theory for strongly continuous semigroups, Math. Ann., Volume 229 (1977) no. 2, pp. 163–171 | DOI | Zbl

[118] Voigt, J. A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatshefte Math., Volume 90 (1980) no. 2, pp. 153–161 | DOI | Zbl

[119] von Foerster; Stohlman, F. H. Some remarks on changing population, The Kinetics of Cell Proliferation, Grune and Stratton, New York, 1959, pp. 382–407

[120] Webb, G.F. Theory of Nonlinear Age-Dependent Population Dynamics, Monogr. Textb. Pure Appl. Math., vol. 89, Marcel Dekker Inc., New York, 1985 | Zbl

[121] Wennberg, B. Stability and exponential convergence for the Boltzmann equation, Arch. Ration. Mech. Anal., Volume 130 (1995) no. 2, pp. 103–144 | DOI | Zbl

[122] Weyl, H. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann., Volume 68 (1910) no. 2, pp. 220–269 | DOI | JFM

[123] Yosida, K. On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Jpn., Volume 1 (1948), pp. 15–21 | DOI | Zbl

[124] Ziff, R.M.; McGrady, E.D. The kinetics of cluster fragmentation and depolymerisation, J. Phys. A, Volume 18 (1985) no. 15, pp. 3027–3037

Cited by Sources: