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Existence results for a Michaud fractional, nonlocal, and randomly position structured fragmentation model. (English) Zbl 1407.35206
Summary: Until now, classical models of clusters’ fission remain unable to fully explain strange phenomena like the phenomenon of shattering [R. M. Ziff and E. D. McGrady, “‘Shattering’ transition in fragmentation”, Phys. Rev. Lett. 58, No. 9, 892–895 (1987)] and the sudden appearance of infinitely many particles in some systems having initial finite number of particles. That is why there is a need to extend classical models to models with fractional derivative order and use new and various techniques to analyze them. In this paper, we prove the existence of strongly continuous solution operators for nonlocal fragmentation models with Michaud time derivative of fractional order [St. G. Samko et al., Fractional integrals and derivatives: theory and applications. Transl. from the Russian. New York, NY: Gordon and Breach (1993; Zbl 0818.26003)]. We focus on the case where the splitting rate is dependent on size and position and where new particles generating from fragmentation are distributed in space randomly according to some probability density. In the analysis, we make use of the substochastic semigroup theory, the subordination principle for differential equations of fractional order [J. Prüss, Evolutionary integral equations and applications. Basel: Birkhäuser Verlag (1993; Zbl 0784.45006); E. G. Bazhlekova, Fract. Calc. Appl. Anal. 3, No. 3, 213–230 (2000; Zbl 1041.34046)], the analogy of Hille-Yosida theorem for fractional model [Prüss, loc. cit.], and useful properties of Mittag-Leffler relaxation function [M. N. Berberan-Santos, J. Math. Chem. 38, No. 4, 629–635 (2005; Zbl 1101.33015)]. We are then able to show that the solution operator to the full model is positive and contractive.

35R11 Fractional partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI
[1] Doungmo Goufo, E. F.; Oukouomi Noutchie, S. C., Global analysis of a discrete non-local and non-autonomous fragmentation dynamics occurring in a moving process, Abstract and Applied Analysis, 2013, (2013) · Zbl 1381.35017
[2] Doungmo Goufo, E. F.; Oukouomi Noutchie, S. C., Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates, Comptes Rendus Mathematique: Comptes Rendus de l’Académie des Sciences I, 351, 19-20, 753-759, (2013) · Zbl 1396.82009
[3] Oukouomi Noutchie, S. C.; Doungmo Goufo, E. F., On the honesty in nonlocal and discrete fragmentation dynamics in size and random position, ISRN Mathematical Analysis, 2013, (2013) · Zbl 1286.35239
[4] Ziff, R. M.; McGrady, E. D., The kinetics of cluster fragmentation and depolymerisation, Journal of Physics A, 18, 15, 3027-3037, (1985)
[5] Garibotti, C. R.; Spiga, G., Boltzmann equation for inelastic scattering, Journal of Physics A, 27, 8, 2709-2717, (1994) · Zbl 0834.45011
[6] Majorana, A.; Milazzo, C., Space homogeneous solutions of the linear semiconductor Boltzmann equation, Journal of Mathematical Analysis and Applications, 259, 2, 609-629, (2001) · Zbl 0986.35112
[7] Wagner, W., Explosion phenomena in stochastic coagulation-fragmentation models, Annals of Applied Probability, 15, 3, 2081-2112, (2005) · Zbl 1082.60075
[8] Oukouomi Noutchie, S. C.; Doungmo Goufo, E. F., Global solvability of a continuous model for nonlocal fragmentation dynamics in a moving medium, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1296.35022
[9] Caputo, M., Linear models of dissipation whose Q is almost frequency independent: part II, Geophysical Journal International, 13, 5, 529-539, (1967)
[10] Oldham, K. B.; Spanier, J., The Fractional Calculus, (1999), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0428.26004
[11] Podlubny, I., Fractional Differential Equations, 198, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[12] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, (2003), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[13] Atangana, A.; Botha, J. F., A generalized groundwater flow equation using the concept of variable order derivative, Boundary Value Problems, 2013, article 53, (2013) · Zbl 1291.35206
[14] Atangana, A.; Vermeulen, P. D., Analytical solutions of a space-time fractional derivative of groundwater flow equation, Abstract and Applied Analysis, 2014, (2014)
[15] Ziff, R. M.; McGrady, E. D., ‘Shattering’ transition in fragmentation, Physical Review Letters, 58, 9, 892-895, (1987)
[16] Anderson, W. J., Continuous-Time Markov Chains. An Applications-Oriented Approach, (1991), New York, NY, USA: Springer, New York, NY, USA · Zbl 0731.60067
[17] Norris, J. R., Markov Chains, (1998), Cambridge, UK: Cambridge University Press, Cambridge, UK
[18] Rudnicki, R.; Wieczorek, R., Phytoplankton dynamics: from the behaviour of cells to a transport equation, Mathematical Modelling of Natural Phenomena, 1, 1, 83-100, (2006) · Zbl 1201.92062
[19] Lachowicz, M.; Wrzosek, D., A nonlocal coagulation-fragmentation model, Applicationes Mathematicae, 27, 1, 45-66, (2000) · Zbl 0994.35054
[20] Balakrishnan, A. V., Fractional powers of closed operators and the semigroups generated by them, Pacific Journal of Mathematics, 10, 419-437, (1960) · Zbl 0103.33502
[21] Yosida, K., Functional Analysis, (1980), New York, NY, USA: Springer, New York, NY, USA · Zbl 0152.32102
[22] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Franctional Integrals and Derivatives. Theory and Application, (1993), Amsterdam, The Netherlands: Gordon and Breach Science, Amsterdam, The Netherlands
[23] Hilfer, R., On New Class of Phase Transitions, Random Magnetism and High Temprature Superconductivity, (1994), Singapore: World Scientific, Singapore
[24] Prüss, J., Evolutionary Integral Equations and Applications, (1993), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0793.45014
[25] Gel’fand, I. M.; Shilov, G., Generalized Functions, 1, (1964), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0115.33101
[26] Lions, J. L.; Peetre, J., Sur une classe d’espaces d’interpolation, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 19, 1, 5-68, (1964) · Zbl 0148.11403
[27] Rubin, B., Fractional Integrals and Potentials, (1996), Harlow, UK: Addison-Wesley, Longman, Harlow, UK · Zbl 0864.26004
[28] Westphal, U., Ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren, Teil I: Halbgruppen-erzeuger, Compositio Mathematica, 22, 67-103, (1970) · Zbl 0194.15401
[29] Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), (2000), New York, NY, USA: Springer, New York, NY, USA · Zbl 0952.47036
[30] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0516.47023
[31] Mittag-Leffler, G. M., Sur la nouvelle fonction Eα\((x)\), Comptes Rendus Mathematique: Comptes Rendus de l’Académie des Sciences II, 137, 554-558, (1903) · JFM 34.0435.01
[32] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, 3, (1955), New York, NY, USA: McGraw-Hill, New York, NY, USA · Zbl 0064.06302
[33] Berberan-Santos, M. N., Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry, 38, 4, 629-635, (2005) · Zbl 1101.33015
[34] Gorenflo, R.; Luchko, Y.; Mainardi, F., Analytical properties and applications of the Wright function, Fractional Calculus & Applied Analysis, 2, 4, 383-414, (1999) · Zbl 1027.33006
[35] Wright, E. M., The generalized Bessel function of order greater than one, The Quarterly Journal of Mathematics, 11, 36-48, (1940) · JFM 66.0322.01
[36] Bazhlekova, E. G., Subordination principle for fractional evolution equations, Fractional Calculus & Applied Analysis, 3, 3, 213-230, (2000) · Zbl 1041.34046
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