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A mathematical analysis of fractional fragmentation dynamics with growth. (English) Zbl 1433.35444
Summary: We make use of the theory of strongly continuous solution operators for fractional models together with the subordination principle for fractional evolution equations [E. Bazhlekova, Fract. Calc. Appl. Anal. 3, No. 3, 213–230 (2000; Zbl 1041.34046)] and [J. Prüss, Evolutionary integral equations and applications. Basel: Birkhäuser Verlag (1993; Zbl 0784.45006)] to analyze and show existence results for a fractional fragmentation model with growth characterized by its growth rate $$r$$. Indeed, strange phenomena like the phenomenon of shattering [E. D. Grady and R. M. Ziff, “Shattering” transition in fragmentation. Phys. Rev. Lett. 58, No. 9, 892–895 (1987)] and the sudden appearance of infinite number of particles in some systems with initial finite particles number could not be fully explained by classical models of fragmentation or aggregation. Then, there is an increasing volition to try new approaches and extend classical models to fractional ones. In the growth model, one of the major challenges in the analysis occurs when $$1/r(x)$$ is integrable at $$X_0 \geq 0$$, the minimum size of a cell. We restrict our analysis to the case of integrability of $$r^{-1}$$ at $$x_0$$. This case needs more considerations on the boundary condition, which, in this paper, is the McKendrick-von Foerster renewal condition. In the process, some properties of Mittag-Leffler relaxation function [M. N. Berberan-Santos, J. Math. Chem. 38, No. 4, 629–635 (2005; Zbl 1101.33015)] are exploited to finally prove that there is a positive solution operator to the full model.

##### MSC:
 35R11 Fractional partial differential equations 33E12 Mittag-Leffler functions and generalizations
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##### References:
 [1] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent: part II,” Journal of Royal Astronomical Society, vol. 13, pp. 529-539, 1967. [2] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1999. · Zbl 0292.26011 [3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 [4] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002 [5] E. D. McGrady and R. M. Ziff, ““Shattering” transition in fragmentation,” Physical Review Letters, vol. 58, no. 9, pp. 892-895, 1987. [6] W. J. Anderson, Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications, Springer, New York, NY, USA, 1991. · Zbl 0731.60067 [7] J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, UK, 1998. · Zbl 0938.60058 [8] R. Rudnicki and R. Wieczorek, “Phytoplankton dynamics: from the behaviour of cells to a transport equation,” Mathematical Modelling of Natural Phenomena, vol. 1, no. 1, pp. 83-100, 2006. · Zbl 1201.92062 · doi:10.1051/mmnp:2006005 · eudml:222345 [9] M. Lachowicz and D. Wrzosek, “A nonlocal coagulation-fragmentation model,” Applicationes Mathematicae, vol. 27, no. 1, pp. 45-66, 2000. · Zbl 0994.35054 · eudml:219259 [10] A. V. Balakrishnan, “Fractional powers of closed operators and the semigroups generated by them,” Pacific Journal of Mathematics, vol. 10, pp. 419-437, 1960. · Zbl 0103.33502 · doi:10.2140/pjm.1960.10.419 [11] K. Yosida, Fonctional Analysis, Springer, Berlin, Germany, 6th edition, 1980. [12] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Franctional Integrals and Derivatives. Theory and Application, Gordon and Breach, Amsterdam, The Netherlands, 1993. · Zbl 0818.26003 [13] E. F. Doungmo Goufo and S. C. Oukouomi Noutchie, “Global analysis of a discrete nonlocal and nonautonomous fragmentation dynamics occurring in a moving process,” Abstract and Applied Analysis, vol. 2013, Article ID 484391, 9 pages, 2013. · Zbl 1381.35017 [14] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer, 2006. · Zbl 1097.47038 [15] E. F. Doungmo Goufo and S. C. Oukouomi Noutchie, “Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates,” Comptes Rendus Mathematique, vol. 351, no. 19-20, pp. 753-759, 2013. · Zbl 1396.82009 [16] S. C. Oukouomi Noutchie and E. F. Doungmo Goufo, “On the honesty in nonlocal and discrete fragmentation dynamics in size and random position,” ISRN Mathematical Analysis, vol. 2013, Article ID 908753, 7 pages, 2013. · Zbl 1286.35239 · doi:10.1155/2013/908753 [17] S. C. Oukouomi Noutchie and E. F. Doungmo Goufo, “Global solvability of a continuous model for nonlocal fragmentation dynamics in a moving medium,” Mathematical Problems in Engineering, vol. 2013, Article ID 320750, 8 pages, 2013. · Zbl 1296.35022 · doi:10.1155/2013/320750 [18] C. R. Garibotti and G. Spiga, “Boltzmann equation for inelastic scattering,” Journal of Physics. A. Mathematical and General, vol. 27, no. 8, pp. 2709-2717, 1994. · Zbl 0834.45011 · doi:10.1088/0305-4470/27/8/009 [19] A. Majorana and C. Milazzo, “Space homogeneous solutions of the linear semiconductor Boltzmann equation,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 609-629, 2001. · Zbl 0986.35112 · doi:10.1006/jmaa.2001.7444 [20] R. M. Ziff and E. D. McGrady, “The kinetics of cluster fragmentation and depolymerisation,” Journal of Physics A, vol. 18, no. 15, pp. 3027-3037, 1985. [21] W. Wagner, “Explosion phenomena in stochastic coagulation-fragmentation models,” The Annals of Applied Probability, vol. 15, no. 3, pp. 2081-2112, 2005. · Zbl 1082.60075 · doi:10.1214/105051605000000386 · arxiv:math/0508488 [22] A. S. Ackleh, “Parameter estimation in a structured algal coagulation-fragmentation model,” Nonlinear Analysis: Theory, Methods & Applications, vol. 28, no. 5, pp. 837-854, 1997. · Zbl 0869.35025 · doi:10.1016/0362-546X(95)00195-2 [23] R. Hilfer, “On new class of phase transitions,” in Random Magnetism High-Temperature Superconductivity, p. 85, World Scientific, Singapore, 1994. [24] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1993. · Zbl 0793.45014 [25] G. M. Mittag-Leffler, “Sur la nouvelle fontion E\alpha (x),” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, vol. 137, pp. 554-558, 1903. [26] A. Érdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 3, McGraw-Hill, New York, NY, USA, 1955. · Zbl 0064.06302 [27] M. N. Berberan-Santos, “Properties of the Mittag-Leffler relaxation function,” Journal of Mathematical Chemistry, vol. 38, no. 4, pp. 629-635, 2005. · Zbl 1101.33015 · doi:10.1007/s10910-005-6909-z [28] R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fractional Calculus and Applied Analysis, vol. 2, no. 4, pp. 383-414, 1999. · Zbl 1027.33006 [29] E. M. Wright, “The generalized Bessel function of order greater than one,” The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 11, pp. 36-48, 1940. · Zbl 0023.14101 · doi:10.1093/qmath/os-11.1.36 [30] I. Gel’fand and G. Shilov, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964. · Zbl 0115.33101 [31] J. L. Lions and J. Peetre, “Sur une classe d’espace d’interpolation,” Publications Mathématiques de l’Institut des Hautes Études Scientifiques, vol. 19, pp. 5-68, 1964. · Zbl 0148.11403 [32] B. Rubin, Fractional Integrals and Potentials, Addison Wesley Longman, Harlow, UK, 1996. · Zbl 0864.26004 [33] U. Westphal, “ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren, Teil I: Halbgruppen-erzeuger,” Compositio Mathematica, vol. 22, pp. 67-103, 1970. · Zbl 0194.15401 · numdam:CM_1970__22_1_67_0 · eudml:89043 [34] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000. · Zbl 0952.47036 · doi:10.1007/b97696 [35] E. G. Bazhlekova, “Subordination principle for fractional evolution equations,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 213-230, 2000. · Zbl 1041.34046 [36] J. Banasiak, S. C. Oukouomi Noutchie, and R. Rudnicki, “Global solvability of a fragmentation-coagulation equation with growth and restricted coagulation,” Journal of Nonlinear Mathematical Physics, vol. 16, supplement 1, pp. 13-26, 2009. · Zbl 1362.92015
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