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On Barenblatt–Zeldovich Intermediate Asymptotics

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Abstract

The concept of intermediate asymptotics for the solution of an evolution equation with initial data and a related solution obtained without initial conditions was introduced by G.N. Barenblatt and Ya.B. Zeldovich in the context of extending the concept of strict determinism in statistical physics and quantum mechanics. Here, according to V.P. Maslov, to axiomatize the mathematical theory, we need to know the conditions satisfied by the initial data of the problem. We show that the correct solvability of a problem without initial conditions for fractional differential equations in a Banach space is a necessary, but not sufficient, condition for intermediate asymptotics. Examples of intermediate asymptotics are given.

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REFERENCES

  1. G. I. Barenblatt and Ya. B. Zeldovich, “Intermediate asymptotics in mathematical physics,” Russ. Math. Surv. 26 (2), 45–61 (1971).

    Article  MathSciNet  Google Scholar 

  2. Ya. B. Zeldovich and D. D. Sokolov, “Fractals, similarity, intermediate asymptotics,” Sov. Phys. Usp. 28, 608–616 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  3. K. Yosida, Functional Analysis (Springer-Verlag, Berlin, 1965).

    Book  Google Scholar 

  4. S. G. Krein, Linear Differential Equations in Banach Spaces (Nauka, Moscow, 1967; Birkhäuser, Boston, 1982).

  5. V. P. Maslov, Asymptotic Methods and Perturbation Theory (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  6. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order: Theory and Applications (Nauka i Tekhnika, Minsk, 1987; Gordon and Breach, London, 1993).

  7. V. V. Uchaikin, Method of Fractional Derivatives (Logos, Ulyanovsk, 2002) [in Russian].

    Google Scholar 

  8. V. E. Fedorov and M. M. Turov, “The defect of a Cauchy type problem for linear equations with several Riemann–Liouville derivatives,” Sib. Math. J. 62 (5), 925–942 (2021).

    Article  MathSciNet  Google Scholar 

  9. G. Da Prato and P. Grisvard, J. Math. Pures Appl. Ser. 9 54, 305–387 (1975).

    Google Scholar 

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Funding

This research was supported by the Russian Science Foundation, subject no. 22-71-10008.

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Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    V. A. Kostin, D. V. Kostin & A. V. Kostin

  2. Voronezh State Pedagogical University, Voronezh, Russia

    D. V. Kostin

Authors
  1. V. A. Kostin
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  2. D. V. Kostin
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  3. A. V. Kostin
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Correspondence to V. A. Kostin, D. V. Kostin or A. V. Kostin.

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Translated by I. Ruzanova

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Kostin, V.A., Kostin, D.V. & Kostin, A.V. On Barenblatt–Zeldovich Intermediate Asymptotics. Dokl. Math. 108, 454–458 (2023). https://doi.org/10.1134/S1064562423701351

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  • DOI: https://doi.org/10.1134/S1064562423701351

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