On Solvability of Boundary Value Problems for Kinetic Operator-Differential Equations

详细信息    查看全文

• 作者：Sergey Pyatkov (1)
  Sergey Popov (2)
  Vasilii Antipin (2)
  
• 关键词：Primary 34G10 ; Secondary 47A50 ; 74A25 ; 82C40 ; Kinetic equation ; Operator ; differential equation ; Krein space ; Forward–backward parabolic equation
• 刊名：Integral Equations and Operator Theory
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：80
• 期：4
• 页码：557-580
• 全文大小：386 KB
• 参考文献：1. Amann, H.: Linear and Quasilinear Parabolic Problems, Birkh?user Verlag, Basel, 1 (1995)
  2. Azizov T. Ya., Iokhvidov I.S.: Linear Operators in Spaces with an Indefinite Metric. John Wiley and Sons, New York (1989)
  3. Beals R.: An abstract treatment of some forward–backward problems of transport and scattering. J. Funct. Anal. 34, 1-0 (1979) CrossRef
  4. Beals R.: Indefinite Sturm–Liouville problems and Half-range completeness. J. Diff. Equ. 56, 391-07 (1985) CrossRef
  5. Beals R., Protopopescu V.: Half-range completeness for the Fokker–Plank equation. J. Stat. Phys. 32, 565-84 (1983) CrossRef
  6. Binding P., Karabash I.M.: Absence of existence and uniqueness for forward–backward parabolic equations on a half-line. Oper. Theory 203, 89-8 (2010) CrossRef
  7. Case K.M., Zweifel P.F.: Linear Transport Theory. Addison-Wesley, Mass (1969)
  8. Cercignani C.: Mathematical Methods in Kinetic Theory. Pergamon Press, New York (1969) CrossRef
  9. Cercignani C.: Theory and Applications of the Boltzmann Equation. Elsevier, New York (1975)
  10. Curgus B.: Boundary value problems in Krein spaces. Glas. Mat. Ser. III 35(55), 45-8 (2000)
  11. Favini A., Yagi A.: Degenerate Differential Equations in Banach Spaces. Marcel Dekker, New York (1999)
  12. Ganchev A., Greenberg W., van der Mee C.V.M.: A class of linear kinetic equations in a Krein space setting. Int. Equat. Oper. Theory 11, 518-35 (1988) CrossRef
  13. Greenberg W., Vander Mee C.V.M., Zweifel P.F.: Generalized kinetic equations. Integr. Equat. Oper. Theory 7, 60-5 (1984) CrossRef
  14. Greenberg W., van der Mee C.V.M., Protopopescu V.: Boundary Value Problems in Abstract Kinetic Theory, Operator Theory, 23. Birkh?user Verlag, Boston (1987) CrossRef
  15. Grisvard P.: Equations differentielles abstraites. Ann. Scient. Ec. Norm. Super. 4 ^{/ e} -Ser 2, 311-95 (1969)
  16. Hangelbroek R.J.: Linear analysis and solution of neutron transport problem. Trans. Theory Statist. Phys. 5, 1-5 (1976) CrossRef
  17. Kaper H.G., Lekkerkerker C.G., Heitmanek J.: Spectral Methods in Linear Transport Theory. Birkhauser Verl, Basel (1982)
  18. Karabash I.M.: Abstract kinetic equations with positive collision operators. Oper. Theory 188, 175-95 (2008)
  19. Kislov N.V.: Nonhomogeneous boundary value problems for differential-operator equations of mixed type, and their applications. Math. Sb. 53, 17-5 (1986) CrossRef
  20. Lar’kin, I.A., Novikov, V.A., Yanenko, N.N.: Nonlinear Equations of Mixed Type. Novosibirsk, Nauka, Russian (1983)
  21. Latrach K.: Compactness properties for linear transport operator with abstract boundary conditions in slab geometry. Transp. Theory Stat. Phys. 22, 39-5 (1993) CrossRef
  22. Latrach K., Mokhtar-Kharroubi M.: Spectral analysis and generation results for streaming operator with multiplying boundary conditions. Positivity 3, 273-96 (1999) CrossRef
  23. Lions, J.L., Magenes, E.: Problemes aux li
• 作者单位：Sergey Pyatkov (1)
  Sergey Popov (2)
  Vasilii Antipin (2)
  
  1. Ugra State University, Chekhov st. 16, 628012, Khanty-Mansiysk, Russia
  2. North-Eastern Federal University, Belinskii st. 58, 677000, Yakutsk, Russia
  
• ISSN：1420-8989

文摘

We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B(t)u t ?em class="a-plus-plus">L(t)u?=?f, where L(t) and B(t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L ?em class="a-plus-plus">λ B is included into one of the half-planes Re λ a or Re λ >?a \({(a\in {\mathbb{R}})}\) . Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions.