NONLINEAR SEMIGROUP APPROACH TO TRANSPORT EQUATIONS WITH DELAYED NEUTRONS

Abstract

Abstract: This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L^p-spaces (1 ≤ p<+∞). Local and non local evolution problems are discussed.

Key words: transport equation with delayed neutrons, general boundary conditions, quasi-accretive operators, mild solution, strong solution, local and global solutions

Abdul-Majeed AL-IZERI, Khalid LATRACH. NONLINEAR SEMIGROUP APPROACH TO TRANSPORT EQUATIONS WITH DELAYED NEUTRONS[J].Acta mathematica scientia,Series B, 2018, 38(6): 1637-1654.

Trendmd

References

[1] Barbu V. Nonlinear Differential Equations of Monotone Types in Banach spaces. Springer, 2010
[2] Bell G I, Glesstone S. Nuclear Reactor Theory. New York:Van Nostrand, 1970
[3] Bénilan Ph. Équations d'Évolution dans un Espace de Banach et Quelconques et Applicá tions. Orsay:Thèse de Doctorat d'Etat, 1972
[4] Bénilan Ph, Crandall M G, Pazy A. Evolution Equations Governed by Accretive Operators. Forthcoming
[5] Browder Felix E. Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull Amer Math Soc, 1967, 7(3):875-882
[6] Cessenat M. Théorèmes de traces L_p pour des espaces de fonctions de la neutronique. C R Acad Sci Paris, Série 1, 1984, 299:831-834
[7] Cessenat M. Théorèmes de traces pour des espaces de fonctions de la neutronique. C R Acad Sci Paris, Série 1, 1985, 300:89-92
[8] De Figueiredo D G, Karlovitz L A. On the radial projection in normed spaces. Bull Amer Math Soc, 1967, 73:364-368
[9] Duderstadt J J, Martin W R. Transport Theory. John Wiley, 1979
[10] Garcia-Falset J. Well-posedness of a nonlinear evolution equation arising in growing cell population. Math Meth Appl Sci, 201134:1658-1666
[11] Garcia-Falset J, Latrach K, Zeghal A. Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations. Nonlinear Anal, 2014, 97:210-227
[12] Greenberg W, Van der Mee C, Protopopescu V. Boundary Value Problems in Abstract Kinetic Theory. Birkhäuser, 1987
[13] Kaper K G. The initial value problem for monoenergetic neutrons in an infinite slab with delayed neutron production. J Math Anal Appl, 1967, 19:207-229
[14] Kaper H G, Lekkerkerker C G, Hejtmanek J. Spectral Methods in Linear Transport Theory. Birkhauser, 1982
[15] Kato T. Nonlinear semigroups and evolution equations. J Math Soc Japan, 1967, 19:508-520
[16] Latrach K. Compactness results for transport equations and applications. Math Models Methods Appl Sci, 200111:1181-1202
[17] Latrach K, Salhi N. Spectral analysis of transport equation with delayed neutrons in slab geometry. Transport Theory Statist Phys (to appear)
[18] Li, Xuezhi, Hao, Yong. The nonlinear transport equation with delayed neutrons. Transport Theory Statist Phys, 1997, 26:153-165
[19] Marti J T. Mathematical foundations of kinetics in neutron transport theory. Nucleonik, 1966, 8:159-163
[20] Mika J. The effect of delayed neutrons on the spectrum of the transport operator. Nucleonik, 1967, 9:46-51
[21] Mokhtar-Kharroubi M. Mathematical Topics in Neutron Transport Theory. New aspects. Series on Advances in Mathematics for Applied Sciences, 46. River Edge, NJ:World Scientific Publishing Co, Inc, 1997
[22] Shanthidevi C N, Matsumoto T, Oharu S. Nonlinear semigroup approach to age structured proliferating cell population with inherited cycle length. Nonlinear Anal Real World Appl, 2008, 9:1905-1917