no. 3
Partial differential equations
A short remark on a growth–fragmentation equation
[Une brève remarque sur une équation de croissance–fragmentation]

Présenté par : Haïm Brézis

Escobedo, Miguel1
1 Departamento de Matemáticas, Universidad del País Vasco (UPV/EHU), 48080 Bilbao, Spain
Comptes Rendus. Mathématique, Tome 355 (2017) no. 3, pp. 290-295.

Nous obtenons une solution explicite d'une équation de croissance–fragmentation avec mesure de dislocation constante. Dans cet exemple, la condition nécessaire sous laquelle les résultats généraux d'existence de solutions globales sont obtenus dans [5] pour le cas dit self-similaire n'est pas satisfaite. La solution est locale et explose en temps fini.

An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in [5] about the existence of global solutions in the so-called self-similar case is not satisfied. The solution is local and blows up in finite time.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.01.013
Escobedo, Miguel 1

1 Departamento de Matemáticas, Universidad del País Vasco (UPV/EHU), 48080 Bilbao, Spain 
@article{CRMATH_2017__355_3_290_0,
     author = {Escobedo, Miguel},
     title = {A short remark on a growth{\textendash}fragmentation equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {290--295},
     publisher = {Elsevier},
     volume = {355},
     number = {3},
     year = {2017},
     doi = {10.1016/j.crma.2017.01.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.01.013/}
}
TY  - JOUR
AU  - Escobedo, Miguel
TI  - A short remark on a growth–fragmentation equation
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 290
EP  - 295
VL  - 355
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2017.01.013/
DO  - 10.1016/j.crma.2017.01.013
LA  - en
ID  - CRMATH_2017__355_3_290_0
ER  - 
%0 Journal Article
%A Escobedo, Miguel
%T A short remark on a growth–fragmentation equation
%J Comptes Rendus. Mathématique
%D 2017
%P 290-295
%V 355
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2017.01.013/
%R 10.1016/j.crma.2017.01.013
%G en
%F CRMATH_2017__355_3_290_0
Escobedo, Miguel. A short remark on a growth–fragmentation equation. Comptes Rendus. Mathématique, Tome 355 (2017) no. 3, pp. 290-295. doi : 10.1016/j.crma.2017.01.013. http://www.numdam.org/articles/10.1016/j.crma.2017.01.013/

[1] Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions, Dover, New York, 1965

[2] Balk, A.M.; Zakharov, V.E. Stability of weak-turbulence Kolmogorov spectra (Zakharov, V.E., ed.), Nonlinear Waves and Weak Turbulence, AMS Translations Series 2, vol. 182, 1998, pp. 1-81

[3] Banasiak, J.; Arlotti, L. Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London Limited, 2006

[4] Bertoin, J.; Stephenson, R. Local explosion in self-similar growth–fragmentation processes, Electron. Commun. Probab., Volume 21 (2016), pp. 21-66

[5] Bertoin, J.; Watson, A.R. Probabilistic aspects of critical growth–fragmentation equations, Adv. Appl. Probab., Volume 48 (2016), pp. 37-61

[6] Bertoin, J.; Curien, N.; Kortchemski, I. Random planar maps & growth-fragmentations (Preprint, available at:) | arXiv

[7] Doumic, M.; Escobedo, M. Time asymptotics for a critical case in fragmentation and growth–fragmentation equations, Kinet. Relat. Models, Volume 9 (2016), pp. 251-297

[8] Doumic, M.; Gabriel, P. Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., Volume 20 (2010), pp. 757-783

[9] M. Escobedo, In preparation.

[10] Misra, O.P.; Lavoine, J.L. Transform Analysis of Generalized Functions, North-Holland Mathematics Studies, Elsevier Science, Amsterdam, New York, Oxford, 1986

[11] Olver, F.W.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010

Cité par Sources :