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On the non existence of non negative solutions to a critical Growth-Fragmentation Equation
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 177-220.

Nous considérons une équation de croissance fragmentation dont les taux de croissance et de fragmentation s’équilibrent et dont le noyau de dislocation est constant. Suivant la valeur de cette constante l’équation vérifie ou non la condition (M + ) introduite par J. Bertoin et A. Watson dans [8]. Nous démontrons que, comme ces auteurs l’avaient suggéré, lorsque la condition n’est pas vérifiée l’équation ne possède pas de solution globale non négative dont les moments satisfont certaines estimations naturelles. Nous montrons également que l’équation peut aussi ne pas avoir de solution locale vérifiant de telles estimations. Lorsqu’une telle solution existe, locale ou globale, une formule explicite est obtenue.

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis (M + ) of J. Bertoin and A. Watson [8] is not necessarily satisfied. It is proved that, as it was first suggested by these authors, when that happens, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exists. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

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DOI : 10.5802/afst.1629
Miguel Escobedo 1

1 Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Miguel Escobedo. On the non existence of non negative solutions to a critical Growth-Fragmentation Equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 177-220. doi : 10.5802/afst.1629. https://afst.centre-mersenne.org/articles/10.5802/afst.1629/

[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables (Milton Abramowitz; Irene A. Stegun, eds.), National Bureau of Standards Applied Mathematics Series, 55, U.S. Department of Commerce, 1964, xiv+1046 pages | MR | Zbl

[2] Alexander M. Balk; Vladimir E. Zakharov Stability of weak-turbulence Kolmogorov spectra, Nonlinear waves and weak turbulence (Translations. Series 2), Volume 182, American Mathematical Society, 1998, pp. 31-81 | MR | Zbl

[3] Jacek Banasiak; Luisa Arlotti Perturbations of positive semigroups with applications, Springer Monographs in Mathematics, Springer, 2006, xiv+438 pages | Zbl

[4] Étienne Bernard; Pierre Gabriel Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate, J. Funct. Anal., Volume 272 (2017) no. 8, pp. 3455-3485 | DOI | MR | Zbl

[5] Jean Bertoin Compensated fragmentation processes and limits of dilated fragmentations, Ann. Probab., Volume 44 (2016) no. 2, pp. 1254-1284 | DOI | MR | Zbl

[6] Jean Bertoin; Timothy Budd; Nicolas Curien; Igor Kortchemski Martingales in self-similar growth-fragmentations and their connections with random planar maps, Probab. Theory Relat. Fields, Volume 172 (2018) no. 3-4, pp. 663-724 | DOI | MR | Zbl

[7] Jean Bertoin; Robin Stephenson Local explosion in self-similar growth-fragmentation processes, Electron. Commun. Probab., Volume 21 (2016), pp. 21-66 | MR | Zbl

[8] Jean Bertoin; Alexander R. Watson Probabilistic aspects of critical growth-fragmentation equations, Adv. Appl. Probab., Volume 48 (2016) no. A, pp. 37-61 | DOI | MR | Zbl

[9] Torsten Carleman Sur la Résolution de Certaines Equations Intégrales, Ark. Mat. Astron. Fys., Volume 16 (1922), pp. 1-19 | Zbl

[10] Marie Doumic Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., Volume 2 (2007) no. 3, pp. 121-152 | DOI | MR | Zbl

[11] Marie Doumic; Miguel Escobedo Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, Volume 9 (2016) no. 2, pp. 251-297 | MR | Zbl

[12] Marie Doumic; Pierre Gabriel Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 5, pp. 757-783 | DOI | MR | Zbl

[13] Marie Doumic; Bruce Van Brunt Explicit Solutions and fine Asymptotics for a critical growth-fragmentation equation (2017) (https://arxiv.org/abs/1704.06087) | Zbl

[14] Miguel Escobedo A short remark on a growth-fragmentation equation, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 3, pp. 290-295 | DOI | MR | Zbl

[15] Miguel Escobedo; Stéphane Mischler; M. Rodriguez Ricard 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

[16] Miguel Escobedo; Juan J. L. Velázquez On the fundamental solution of a linearized homogeneous coagulation equation, Commun. Math. Phys., Volume 297 (2010) no. 3, pp. 759-816 | DOI | MR | Zbl

[17] David A. Kveselava The solution of a boundary problem of the theory of function, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., Volume 53 (1946), pp. 679-682 | MR | Zbl

[18] Jean-François Le Gall Random geometry on the sphere, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1 (2014), pp. 421-442 | Zbl

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

[20] Philippe Michel; Stéphane Mischler; Benoît Perthame General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl., Volume 84 (2005) no. 9, pp. 1235-1260 | DOI | MR | Zbl

[21] Grégory Miermont Aspects of random maps Saint-Flour lecture notes (Preliminary version)

[22] Om P. Misra; J. L. Lavoine Transform analysis of generalized functions, North-Holland Mathematics Studies, 119, North-Holland, 1986, xiv+332 pages | MR | Zbl

[23] NIST handbook of mathematical functions (Frank W. J. Olver; Daniel W. Lozier; Ronald F. Boisvert; Charles W. Clark, eds.), U.S. Department of Commerce; Cambridge University Press, 2010, xvi+951 pages | Zbl

[24] Raymond E. A. C. Paley; Norbert Wiener Fourier transforms in the complex domain, Colloquium Publications, 19, American Mathematical Society, 1987, x+184 pages (reprint of the 1934 original) | MR

[25] Benoît Perthame Transport equations in biology, Frontiers in Mathematics, Birkhäuser, 2007, x+198 pages | Zbl

[26] Robert M. Ziff; Ed D. McGrady The kinetics of cluster fragmentation and depolymerisation, J. Phys. A, Math. Gen., Volume 18 (1985) no. 15, pp. 3027-3037 | DOI | MR

[27] Robert M. Ziff; Ed D. McGrady Kinetics of Polymer Degradation, Macromolecules, Volume 19 (1986), pp. 2513-2519 | DOI

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