Siberian Mathematical Journal

, Volume 28, Issue 6, pp 906–911 | Cite as

Conditions for well-posedness in the Hadamard sense in spaces of generalized functions

  • V. K. Ivanov


We make some remarks in connection with the results of Secs. 6–10.
  1. 1.

    We have used expansions into orthogonal series. In the general case it is natural to use spectral expansions of linear selfadjoint operators. For this one may use the results of [13, 14].

  2. 2.

    Can the concept of weak well-posedness be carried over to nonlinear problems? Here, probably, J. F. Colombeau's method will turn out to be useful (see [13, 14]).



Generalize Function Nonlinear Problem Selfadjoint Operator Orthogonal Series Spectral Expansion 
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Literature Cited

  1. 1.
    S. L. Sobolev, “Methode nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales,” Mat. Sb.,1 (43), 39–72 (1936).Google Scholar
  2. 2.
    A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York (1977).Google Scholar
  3. 3.
    M. M. Lavrent'ev, Conditionally Well-Posed Problems for Differential Equations [in Russian], Novosibirsk State Univ. (1973).Google Scholar
  4. 4.
    V. P. Maslov, “The existence of a solution of an ill-posed problem is equivalent to the convergence of a regularization process,” Usp. Mat. Nauk,23, No. 3, 183–184 (1968).Google Scholar
  5. 5.
    A. H. Zemanian, Generalized Integral Transformations, Interscience, New York (1968).Google Scholar
  6. 6.
    G. N. Mil'shtein, “The extension of semigroups of operators in locally convex spaces,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 91–95 (1977).Google Scholar
  7. 7.
    J. de Graaf, “A theory of generalized functions based on holomorphic semigroups,” Nederl. Akad. Wetensch. Proc., A:86A, No. 4, 407–420 (1983); B:87A, No. 2, 155–171 (1984); C:87A, No. 2, 173–187 (1984).Google Scholar
  8. 8.
    S. Pilipovic, “Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions,” SIAM J. Math. Anal.,17, No. 2, 477–484 (1986).Google Scholar
  9. 9.
    V. Wrobel, “Generating Frechet-Montel spaces that are not Schwartz by closed linear operators,” Arch. Math. (Basel),46, No. 3, 257–260 (1986).Google Scholar
  10. 10.
    A. Szaz, “Periodic generalized functions,” Publ. Math. Debrecen,25, No. 3–4, 229–235 (1978).Google Scholar
  11. 11.
    A. Szaz, “Generalized periodic distributions,” Rev. Roumaine Math. Pures Appl.,23, No. 10, 1577–1582 (1978).Google Scholar
  12. 12.
    Yu. A. Dubinskii, “The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physics,” Usp. Mat. Nauk,37, No. 5, 97–137 (1982).Google Scholar
  13. 13.
    I. M. Gel'fand and A. K. Kostyuchenko, “Expansion in eigenfunctions of differential and other operators,” Dokl. Akad. Nauk SSSR,103, No. 3, 349–352 (1955).Google Scholar
  14. 14.
    I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 3: Theory of Differential Equations, Academic Press, New York (1967).Google Scholar
  15. 15.
    J. F. Colombeau, “A multiplication of distributions,” J. Math. Anal. Appl.,94, No. 1, 96–115 (1983).Google Scholar
  16. 16.
    J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland Math. Studies, Vol. 113, North-Holland, Amsterdam (1985).Google Scholar

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© Plenum Publishing Corporation 1988

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  • V. K. Ivanov

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