Skip to main content
SpringerLink
Log in

Peculiarities and regularization of ill-posed Cauchy problems with differential operators

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. A. B. Antonevich and Ya. V. Radyno, Functional Analysis and Integral Equations [in Russian], Publ. “Universitetskoe,” Minsk, 1984.

    MATH  Google Scholar 

  2. U. A. Anufrieva, “Investigation of ill-posed differential-operator problems by semigroup technics,” Dissertation for Candidate of Physicomathematical Sciences, Ekaterinburg, 1999.

  3. U. A. Anufrieva “Semigroups connected with systems correct by Petrovsky,” Spectral and Evolution Problems, Proc. of the Fifteenth Crimean Autumn Mathematical School-Symposium, 18–29 Sept. 2004, Simferopol, 15, 103–108 (2005).

    Google Scholar 

  4. W. Arendt, “Vector valued Laplace transforms and Cauchy problems,” Isr. J. Math., 59, No. 3, 327–352 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Arendt, O. El-Mennaoui, and V. Keyantuo, “Local integrated semigroups: evolution with jumps of regularity,” J. Math. Anal. Appl., 186, No. 2, 572–595 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Arendt, Ch. Batty, et al., Vector-Valued Laplace Transform and Cauchy Problems, Birkhäuser, Basel-Boston-Berlin, 2001.

    Google Scholar 

  7. A. V. Balakrishnan, Applied Functional Analysis, Springer, New York-Heidelberg-Berlin, 1976.

    MATH  Google Scholar 

  8. A. G. Baskakov and K. I. Chernyshov, “Spectral analysis of linear relations and degenerate semigroups of operators,” Mat. Sb., 193, No. 11, 3–24 (2002).

    MathSciNet  Google Scholar 

  9. H. Bremermann, Distributions, Complex Variables and Fourier Transforms, Addison-Wesley, 1965.

  10. J. Chazarain, “Problemes de Cauchy au sens des distributions vectorielles et applications,” C. R. Acad. Sci., Paris, 266, 10–13 (1968).

    MATH  MathSciNet  Google Scholar 

  11. J. Chazarain, “Problemes de Cauchy abstraits et applications a quelques problemes mixtes,” J. Funct. Anal., 7, No. 3, 386–446 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Cioranescu and G. Lumer, “Regularization of evolution equations via kernels K(t), K-evolution operators and convoluted semigroups, generation theorems,” Seminar Notes in Funct. Anal. and PDEs, 1993–1994, Louisiana State Univ., Baton Rouge, 45–52 (1994).

    Google Scholar 

  13. I. Cioranescu, “Local convoluted semigroups,” Lect. Notes Pure Appl. Math., 168, 107–122 (1995).

    MathSciNet  Google Scholar 

  14. G. Da Prato, “Semigruppi regolarizzabili,” Ric. Mat., 15, 223–248 (1966).

    MATH  Google Scholar 

  15. E. B. Davis and M. M. Pang, “The Cauchy problem and a generalization of the Hille-Yosida theorem,” Proc. London Math. Soc., 55, 181–208 (1987).

    Article  MathSciNet  Google Scholar 

  16. R. de Laubenfels, Existence Families, Functional Calculi and Evolution Equations, Springer, 1994.

  17. A. A. Desin, Partial Differential Equations: an Introduction to a General Theory of Linear Boundary Value Problems, Springer, New York-Berlin, 1987.

    Google Scholar 

  18. N. Dunford and J. T. Schwartz, Linear Operators, Vol. 1–3, Interscience, New York, 1958–1971.

    MATH  Google Scholar 

  19. K.-J. Engel and R. Nagel, “One-parameter semigroups for linear evolution equations,” Grad. Texts Math., 194 (1999).

  20. H. O. Fattorini, “The Cauchy problem,” Encycl. Math. Appl., 18 (1983).

  21. I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1, Generalezed Functions and Operations with Them, Academic Press, New York, 1966.

    Google Scholar 

  22. I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 2, Spaces of Test and Generalized Functions, Academic Press, New York, 1967.

    Google Scholar 

  23. I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 3, Some Questions of the Theory of Differential Equations, Academic Press, New York, 1968.

    Google Scholar 

  24. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford, New York, 1985.

    MATH  Google Scholar 

  25. E. Hille and R. S. Phillips, “Functional analysis and semi-groups,” Am. Math. Soc. Coll. Publ., 31 (1957).

  26. V. K. Ivanov and I. V. Melnikova, “New generalized functions and weak well-posedness of operator problems,” Dokl. Akad. Nauk SSSR, 317, No. 1, 22–26 (1991); English transl.: Soviet Math. Dokl., 43 (1991).

    MathSciNet  Google Scholar 

  27. V. K. Ivanov, I. V. Melnikova, and A. I. Filinkov, Differential-Operator Equations and Ill-Posed Problems [in Russian], Nauka, Moscow, 1995.

    Google Scholar 

  28. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications [in Russian], Nauka, Moscow, 1978.

    MATH  Google Scholar 

  29. A. Kaminski, D. Kovacevic, and S. Pilipovic, “The equivalence of various definitions of the convolution of ultradistributions,” Proc. Math. Inst. RAS, 203, 307–322 (1994).

    Google Scholar 

  30. T. Kato, Pertubation Theory for Linear Operators, Springer, Berlin, 1980.

    Google Scholar 

  31. W. Ketc and P. P. Teodorescu, Introducere in Teoria Distributilor cu Aplicati in Tehnică, Editura Tehnică, Buguresti, 1975.

    Google Scholar 

  32. H. Kellermann and M. Hieber, “Integrated semigroups,” Funct. Anal., 84, 160–180 (1989).

    Article  MathSciNet  Google Scholar 

  33. A. N. Kolmogorov and S. V. Fomin, Elements of Theory of Functions and of Functional Analysis [in Russian], Nauka, Moscow, 1968.

    Google Scholar 

  34. H. Komatsu, “Ultradistributions, I. Structure theorems and characterization,” J. Fac. Sci. Univ. Tokyo, 20, No. 1, 25–106 (1973).

    MATH  MathSciNet  Google Scholar 

  35. S. G. Krein, Linear Differential Equations in a Banach Space, Am. Math. Soc., Providence, R. I., 1972.

    Google Scholar 

  36. S. G. Krein and M. I. Hazan, “Differential equations in a Banach space,” Itogi Nauki Tekh., Mat. Anal., 21, 130–232 (1983).

    Google Scholar 

  37. R. Lattes and J.-L. Lions, The Method of Quasi-Reversibility: Applications to Partial Differential Equations, American Elsevier, New York, 1969.

    Google Scholar 

  38. M. M. Lavrentev, Conditionally Well-Posed Problems for Differential Equations [in Russian], VC SO AN SSSR, Novosibirsk, 1973.

    Google Scholar 

  39. J.-L. Lions, “Les semi-groupes distributions,” Port. Math., 19, No. 3–4, 141–161 (1960).

    MATH  Google Scholar 

  40. Yu. I. Lyubich, “The classical and local Laplace transformation in an abstract Cauchy problem,” Russ. Math. Surv., 21, No. 3 (129), 3–51 (1966).

    Article  MATH  Google Scholar 

  41. G. Lyzik, “On extendible ultradistributions,” Bull. Pol. Acad. Sci., Math., 43, No. 1, 29–40 (1995).

    Google Scholar 

  42. S. Mandelbrojt, “Fonctions entières et transformées de Fourier, Applications,” Publ. Math. Soc. Japan, Tokyo, No. 10 (1967).

  43. M. V. Markin, “On the ultradifferentiability of weak solutions of a first-order operator-differential equation in a Hlibert space,” Dokl. Akad. Nauk Ukr., No. 6, 22–26 (1996).

    Google Scholar 

  44. I. V. Melnikova, “Well-posed and ill-posed problems for differential-operator equations in Banach spaces,” Dissertation for Doctor of Physicomathematical Sciences, Novosibirsk, 1988 [in Russian].

  45. I. V. Melnikova, “General theory of ill-posed Cauchy problem,” J. Inverse Ill-Posed Probl., 2, No. 2 (1995).

  46. I. V. Melnikova, “Properties of Lions’s d-semigroups and generalized well-posedness of the Cauchy problem,” Funct. Anal. Prilozh., 31, No. 3, 23–37 (1997).

    MathSciNet  Google Scholar 

  47. I. V. Melnikova, “Well-posedness of differential-operator problems I: the Cauchy problem in spaces of distributions,” Itogi Nauki Tekh., Ser. Sovr. Mat. Prilozh., Analiz, English transl.: J. Math. Sci. (1998).

  48. I. V. Melnikova, “Cauchy problem for inclusion in Banach spaces and spaces of distributions,” Sib. Mat. Zh., 42, No. 4, 892–910 (2001).

    MathSciNet  Google Scholar 

  49. I. V. Melnikova, “Semigroup regularization of differential problems,” Dokl. Akad. Nauk, 393, No. 6, 1–5 (2003).

    MathSciNet  Google Scholar 

  50. I. V. Melnikova, “Semigroup regularization for ill-posed Cauchy problems,” in: Proc. Second Int. Conf. “Semigroup of operators: theory and applications, 10–14 Sept. 2001, Rio de Janeiro, 189–199 (2002).

  51. I. V. Melnikova, “About regularization in a broad sense,” in: Spectral and Evolution Problems, Proc. Fourteenth Crimean Autumn Math. School-Symposium, 18–29 Sept., 2003, Simferopol, 15, 192–203 (2005).

    Google Scholar 

  52. I. V. Melnikova and M. A. Al’shansky, “Well-posedness of the Cauchy problem in a Banach space: regular and degenerate cases,” Itogi Nauki Tekhn., Ser. Sovr. Mat. Prilozh., Analiz, 27, 5–64 (1995); English transl.: J. Math. Sci., 87, No. 4, 3732–3780 (1998).

    Google Scholar 

  53. I. V. Melnikova, U. A. Anufrieva, and A. I. Filinkov, “Laplace transform of K-semigroups and well-posedness of the Cauchy problems,” J. Int. Trans. Spec. Funct., 8, No. 1–2, 1–20 (1999).

    Google Scholar 

  54. I. V. Melnikova and S. V. Bochkareva, “C-semigroups and regularization of an ill-posed Cauchy problem,” Dokl. Akad. Nauk, 329, No. 3, 270–273 (1993); English transl.: Russian Acad. Sci. Dokl. Math., 47 (1993).

    MathSciNet  Google Scholar 

  55. I. V. Melnikova and A. I. Filinkov, “Integrated semigroups and C-semigroups. Well-posedness and regularization of differential-operator problems,” Russ. Math. Surv., 49, No. 6, 111–150 (1994).

    MathSciNet  Google Scholar 

  56. I. V. Melnikova and A. I. Filinkov, “The Cauchy problem: three approaches,” in: Monographs and Surveys in Pure and Applied Mathematics, Vol. 120, Chapman&Hall, London-New York, 2001.

    Google Scholar 

  57. I. V. Melnikova, Q. Zheng, and J. Zhang, “Regularization of weakly ill-posed Cauchy problems,” J. Inverse Ill-Posed Probl., 10, No. 5 (2002).

    Google Scholar 

  58. S. Mizohata, Theory of Partial Differential Equations [in Russian], Mir, Moscow, 1977.

    Google Scholar 

  59. I. Miyadera and N. Tanaka, “Exponential R-semigroups and generation of semigroups,” J. Math. Anal. Appl., No. 2, 358–365 (1989).

  60. F. Neubrander, “Integrated semigroups and their applications to the abstract Cauchy problem,” Pacific J. Math., 135, No. 1, 111–155 (1988).

    MATH  MathSciNet  Google Scholar 

  61. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin-New York, 1983.

    MATH  Google Scholar 

  62. S. Philipovič, “Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions,” SIAM J. Math. Anal., 17, 477–484 (1986).

    Article  MathSciNet  Google Scholar 

  63. R. S. Phillips, “Inversion formula for Laplace transforms and semi-groups of linear operators,” Ann. Math., 59, 325–356 (1954).

    Article  Google Scholar 

  64. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York-London, 1972.

    Google Scholar 

  65. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-San Francisco-London, 1975.

    MATH  Google Scholar 

  66. R. D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. 1, Springer, New York-Heidelberg-Berlin, 1978.

    MATH  Google Scholar 

  67. R. D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. 2, Springer, New York-Heidelberg-Berlin, 1981.

    MATH  Google Scholar 

  68. A. P. Robertson and V. Zh. Robertson, Topological Vector Spaces [in Russian], Mir, Moscow, 1967.

    MATH  Google Scholar 

  69. L. Schwartz, “Théorie des distriutions à valeurs vectorielles, 1-ere partie,” Ann. Inst. Fourier, 7, 1–144 (1957).

    MATH  Google Scholar 

  70. L. Schwartz, “Théorie des distriutions à valeurs vectorielles, 2-ere partie,” Ann. Inst. Fourier, 8, 1–207 (1958).

    MATH  Google Scholar 

  71. P. E. Sobolevskii, “On semigroups of growth order α,” Dokl. Akad. Nauk SSSR, 196, No. 3, 535–537 (1971).

    MathSciNet  Google Scholar 

  72. M. Sova, “Cosine operator functions,” Rozpr. Math., 49, 1–47 (1966).

    MathSciNet  Google Scholar 

  73. N. Tanaka and I. Miyadera, “Some remarks on C-semigroups and integrated semigroups,” Proc. Japan Acad., 63, No. 5, 139–142 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  74. N. Tanaka and N. Okazawa, “Local C-semigroups and local integrated semigroups,” Proc. London Math. Soc., 61, No. 3, 63–90 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  75. A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow, 1979.

    Google Scholar 

  76. V. V. Vasiliev, S. G. Krein, and S. I. Piskarev, “Semigroups of operators, cosine operator-valued functions and linear differential equations,” Itogi Nauki Tekh., Mat. Anal., 28, 87–202 (1990).

    Google Scholar 

  77. D. V. Widder, An Introduction to Laplace Transform Theory, Academic Press, New York, 1971.

    Google Scholar 

  78. K. Yosida, Functional Analysis, Springer, Berlin-New York, 1980.

    MATH  Google Scholar 

  79. S. Zaidman, “Functional analysis and differential equations in abstract spaces,” Monographs and Surveys in Pure and Applied Mathematics, Vol. 100, Chapman&Hall/CRC, Boca Raton-London-New York, 1999.

    Google Scholar 

  80. A. G. Zemanian, Generalized Integral Transformations, Interscience Publishers, New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Department of Mathematical Analysis and Theory of Functions, Ural State University, Russia

    I. V. Melnikova & U. A. Anufrieva

Authors
  1. I. V. Melnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. U. A. Anufrieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Melnikova.

Additional information

__________

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 14, Differential Equations and Theory of Semigroups, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Melnikova, I.V., Anufrieva, U.A. Peculiarities and regularization of ill-posed Cauchy problems with differential operators. J Math Sci 148, 481–632 (2008). https://doi.org/10.1007/s10958-008-0012-5

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-0012-5

Keywords

Search

Navigation