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March  2012, 5(1): 113-128. doi: 10.3934/krm.2012.5.113

Estimates of solutions of linear neutron transport equation at large time and spectral singularities

Roman Romanov 1,

1. 

Laboratory of Quantum Networks and Department of Mathematical Physic, Faculty of Physics, St.Petersburg State University, 198504, Saint Petersburg, Russian Federation

Received  August 2010 Revised  August 2011 Published  January 2012

The spectral analysis of a dissipative linear transport operator with a polynomial collision integral by the Szőkefalvi-Nagy - Foiaş functional model is given. An exact estimate for the remainder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has at most finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum. An upper estimate for the remainder is established.
Keywords: Transport equation, spectral singularities, asymptotics of semigroups..
Mathematics Subject Classification: Primary: 47A45; Secondary: 35Q2.
Citation: Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic & Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113
References:
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show all references

References:
[1]

(French) [Harmonic Analysis of Operators in Hilbert Space], Masson et Cie, Paris, Académiai Kiadó, Budapest, 1967.  Google Scholar

[2]

J. Math. Mech., 11 (1962), 173-181.  Google Scholar

[3]

Comm. Pure Appl. Math., 8 (1955), 217-234. doi: 10.1002/cpa.3160080202.  Google Scholar

[4]

Duke Math. J., 23 (1956), 125-142. doi: 10.1215/S0012-7094-56-02312-2.  Google Scholar

[5]

Indiana Univ. Math. J., 51 (2002), 1389-1425. doi: 10.1512/iumj.2002.51.2180.  Google Scholar

[6]

Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 123-148, 240.  Google Scholar

[7]

(Russian) Trudy Mosk. Mat. Obšč., 3 (1954), 181-270; English transl., Amer. Math. Soc. Transl. (2), 16 (1960), 103-193.  Google Scholar

[8]

in "Partial Differential Equations, VIII" (ed. M. Shubin), Encyclopaedia Math. Sci., 65, Springer, Berlin, (1996), 87-153.  Google Scholar

[9]

(Russian) Boundary value problems of mathematical physics, 10, Trudy Mat. Inst. Steklov, 147 (1980), 86-114, 203; English transl., Proc. Steklov Inst. Math., 147 (1981), 85-116.  Google Scholar

[10]

(Russian) in "Wave Propagation. Scattering Theory" (ed. M. Birman), LGU, Leningrad, (1987), 132-155; English transl., Amer. Math. Soc. Transl. Ser. 2, 157, AMS, Providence, RI, (1993), 127-149.  Google Scholar

[11]

Acta Sci. Math. (Szeged), 70 (2004), 379-403.  Google Scholar

[12]

With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 273, Springer-Verlag, Berlin, 1986.  Google Scholar

[13]

in "Spectral Methods for Operators of Mathematical Physics" (Bedlewo, 2002), Oper. Theory Adv. Appl., 154, Birkhäuser, Basel, (2004), 179-184.  Google Scholar

[14]

(Russian) Trudy Moskov. Mat. Obšč., 19 (1968), 211-270.  Google Scholar

[15]

(Russian) Algebra i Analiz, 14 (2002), 158-195; English transl., St. Petersburg Math. J., 14 (2003), 655-682.  Google Scholar

[16]

Math. Notes, 89 (2011), 106-116. doi: 10.1134/S0001434611010111.  Google Scholar

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