Advertisement

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

  • V. Zh. Sakbaeva
Article

Abstract

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.

Keywords

Banach Space Cauchy Problem STEKLOV Institute Linear Manifold Contraction Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].MATHGoogle Scholar
  2. 2.
    V. V. Zhikov, “To the Problem of Passage to the Limit in Divergent Nonuniformly Elliptic Equations,” Funkts. Anal. Prilozh. 35(1), 23–39 (2001) [Funct. Anal. Appl. 35, 19–33 (2001)].MathSciNetGoogle Scholar
  3. 3.
    A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1986) [in Russian].Google Scholar
  4. 4.
    O. A. Oleinik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form (VINITI, Moscow, 1971; Plenum, New York, 1973), Itogi Nauki, Ser.: Mat., Mat. Anal. 1969.Google Scholar
  5. 5.
    P. I. Plotnikov and S. A. Sazhenkov, “Cauchy Problem for the Graetz-Nusselt Ultraparabolic Equation,” Dokl. Akad. Nauk 401(4), 455–458 (2005) [Dokl. Math. 71 (2), 234–237 (2005)].MathSciNetGoogle Scholar
  6. 6.
    L. V. Korobenko and V. Zh. Sakbaev, “Solution of the Diffusion Equation with Discontinuous Coefficient on a Straight Line,” in Some Problems of Fundamental and Applied Mathematics (Moscow Inst. Phys. Technol., Moscow, 2006), pp. 71–85 [in Russian].Google Scholar
  7. 7.
    O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996) [in Russian].Google Scholar
  8. 8.
    L. N. Slobodetskii, “Generalized Sobolev Spaces and Their Application to Boundary Problems for Partial Differential Equations,” Uch. Zap. Leningr. Gos. Pedagog. Inst. 197, 54–112 (1958) [Am. Math. Soc. Transl., Ser. 2, 57, 207–275 (1966)].Google Scholar
  9. 9.
    S. G. Krein, Linear Differential Equations in Banach Space (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1972).Google Scholar
  10. 10.
    K. Yosida, Functional Analysis (Springer, Berlin, 1965; Mir, Moscow, 1967).MATHGoogle Scholar
  11. 11.
    V. Zh. Sakbaev, “On Properties of Solutions to the Cauchy Problem for a Schrödinger Equation Degenerate outside an Interval and Spectral Aspects of Regularization,” Sovrem. Mat., Fundam. Napravl. 21, 87–113 (2007).MathSciNetGoogle Scholar
  12. 12.
    V. Zh. Sakbaev, “Degeneration and Regularization of the Operator in the Cauchy Problem for the Schrödinger Equation,” Sovrem. Mat. Prilozh. 38, 95–109 (2006) [J. Math. Sci. 147 (1), 6483–6497 (2007)].Google Scholar
  13. 13.
    V. Zh. Sakbaev, “Functionals on Solutions of the Cauchy Problem for the Schrödinger Equation Degenerate on a Half-Line,” Zh. Vychisl. Mat. Mat. Fiz. 44(9), 1654–1673 (2004) [Comput. Math. Math. Phys. 44, 1573–1591 (2004)].MATHMathSciNetGoogle Scholar
  14. 14.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972; Mir, Moscow, 1977).Google Scholar
  15. 15.
    Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups (North-Holland, Amsterdam, 1987; Mir, Moscow, 1992).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Department of Higher MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyi, Moscow oblastRussia

Personalised recommendations