Advertisement

Journal of Mathematical Sciences

, Volume 156, Issue 5, pp 799–812 | Cite as

On analytic solutions of the heat equation with an operator coefficient

  • A. Vershynina
  • S. Gefter
Article

Let A be a bounded linear operator on a Banach space and let g a be vector-valued function that is analytic in a neighborhood of the origin of ℝ. We obtain conditions of the existence of analytic solutions for the Cauchy problem \( \left\{ {\begin{array}{l} {\frac{{\partial u}}{{\partial t}} = A^2 \frac{{\partial ^2 u}}{{\partial x^2 }},} \\ {u\left( {0,x} \right) = g\left( x \right).} \\ \end{array} } \right. \) Moreover, we consider a representation of the solution of this problem as a Poisson integral and study the Cauchy problem for the corresponding inhomogeneous equation. Bibliography: 22 titles.

Keywords

Banach Space Linear Operator Cauchy Problem Heat Equation Bounded Linear Operator 
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.
    E. Hille, Ordinary Differential Equations in the Complex Domain, A Wiley-Interscience publication, New York-London (1976).zbMATHGoogle Scholar
  2. 2.
    J. Hadamar, Le Problème de Cauchy et les Équations aux Derivees Partielles Linéaires Hyperboliques, Hermann, Paris (1932).Google Scholar
  3. 3.
    R. Courant, Partial Differential Equations, New York-London (1962).Google Scholar
  4. 4.
    I. Petrovskii, Partial Differential Equations, Iliffe Books Ltd., London (1967).Google Scholar
  5. 5.
    S. V. Kowalewsky, “Zur Theorie der partiellen Differentialgleichungen,” J. reine angew. Math., 80, 1–32 (1875).Google Scholar
  6. 6.
    C. Riquie, Les Systémes d'Èquations aux Dérivées Partielles, Gauthier-Villars, Paris (1910).Google Scholar
  7. 7.
    P. Rosebloom, “The majorant method. Partial differential equations,” in: Proc. of the Fourth Symposium in Pure Mathemetics, Amer. Math. Soc. (1961), pp. 51–72.Google Scholar
  8. 8.
    V. Palamodov, “Differential operators on the class of convergent power series and the Weierstrass auxiliary lemma,” Funkts. Analiz Prilozh., 2, 235–244 (1968).CrossRefMathSciNetGoogle Scholar
  9. 9.
    F. Treves, “An abstract nonlinear Cauchy-Kovalevskaya theorem,” Trans. Amer. Math. Soc., 150, 77–92 (1970).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    S. Mizohata, “On Kowalewskian systems,” Russ. Math. Surv. 29, 223–235 (1974).CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    L. Ovsyannikov, “Abstract form of the Cauchy-Kovalevskaya theorem and its applications,” Partial Diff. Eqs, Novosibirsk, 88–94 (1980).Google Scholar
  12. 12.
    S. Öuchi, “Characteristic Cauchy problems and solutions of formal power series,” Ann. Inst. Fourier, 33, 131–176 (1983).Google Scholar
  13. 13.
    D. Lutz, M. Miyake, and R. Schafke, “On the Borel summability of divergent solutions of the heat equation,” Nagoya Math. J., 154, 1–29 (1999).MathSciNetzbMATHGoogle Scholar
  14. 14.
    W. Balser, “Divergent solutions of the heat equation: on an article of Lutz, Miyake, and Schafke, ” Pacific J. Math., 188, 53–63 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Gorbachuk, “On the well-posedness of the Cauchy problem for operator-differential equations in classes of analytic vector functions,” Dokl. Ros. Akad. Nauk, 374, 7–9 (2000).MathSciNetGoogle Scholar
  16. 16.
    M. Gorbachuk, “An operator approach to the Cauchy-Kovalevskaya theorem,” J. Math. Sci., 5, 1527–1532 (2000).CrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Chulkov, “On the convergence of formal solutions of systems of partial differential equations, ” Funkts. Analiz Prilozh., 39, 215–224 (2005).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    A. Khovanskii and S. Chulkov, “The Hilbert polynomial for systems of linear partial differential equations with analytic coefficients,” Izv. Math., 70, 153–169 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    E. Hille and R. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Coll. Publ., 31, Providence (1957).Google Scholar
  20. 20.
    H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer-Verlag (1971).Google Scholar
  21. 21.
    Ju. Dalec'kii and M. Krein, Stability of Differential Equations in a Banach Space, Amer. Math. Soc, Providence (1974).Google Scholar
  22. 22.
    S. Gefter and V. Mokrenyuk, “The power series {ie812-01} and holomorphic solutions of some differential equations in a Banach space,” J. Math. Phys., Anal., Geom., 1, 53–70 (2005).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.V. N. Karazin Kharkiv National UniversityKharkivUkraine

Personalised recommendations