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Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

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Abstract

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.

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References

  1. Lyakhov, D.A. and Lomovtsev, F.E., On Weak Solutions of the Cauchy Problem for a Second-Order Hyperbolic Operator-Differential Equation with a Variable Domain, Dokl. Nats. Akad. Nauk Belarusi, 2010, vol. 54, no. 1, pp. 44–49.

    MATH  MathSciNet  Google Scholar 

  2. Lyakhov, D.A. and Lomovtsev, F.E., Method of Weak Solutions of Auxiliary Cauchy Problem for the Study of the Smoothness of Solutions of Second-Order Hyperbolic Differential-Operator Equations with Variable Domains, Vestn. Belarus. Gos. Univ. Ser. 1, 2010, no. 2, pp. 9–14.

    Google Scholar 

  3. Lomovtsev, F.E., Differentiation and Integration with Respect to a Parameter of Unbounded Variable Operators with Variable Domains, Dokl. Nats. Akad. Nauk Belarusi, 1999, vol. 43, no. 1, pp. 13–15.

    MATH  MathSciNet  Google Scholar 

  4. Lomovtsev, F.E., Second-Order Hyperbolic Differential Equations with Discontinuous Operator Coefficients, Differ. Uravn., 1997, vol. 33, no. 10, pp. 1394–1403.

    MathSciNet  Google Scholar 

  5. Lomovtsev, F.E., Second-Order Hyperbolic Operator-Differential Equations with Variable Domains of Discontinuous Operator Coefficients, Dokl. Nats. Akad. Nauk Belarusi, 2001, vol. 45, no. 3, pp. 37–40.

    MATH  MathSciNet  Google Scholar 

  6. Lomovtsev, F.E., Necessary and Sufficient Conditions for the Unique Solvability of the Cauchy Problem for Second-Order Hyperbolic Differential Equations with a Variable Domain of Operator Coefficients, Differ. Uravn., 1992, vol. 28, no. 5, pp. 873–886.

    MATH  MathSciNet  Google Scholar 

  7. Lomovtsev, F.E., Second-Order Hyperbolic Operator-Differential Equations with Variable Domains of Smooth Operator Coefficients, Dokl. Nats. Akad. Nauk Belarusi, 2001, vol. 45, no. 1, pp. 34–37.

    MATH  MathSciNet  Google Scholar 

  8. Lomovtsev, F.E., Cauchy Problems for Quasihyperbolic Factorized Differential Equations with Variable Domains of Discontinuous Operators, Differ. Uravn., 2007, vol. 43, no. 9, pp. 1433–1436.

    MathSciNet  Google Scholar 

  9. Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.

    Google Scholar 

  10. Treves, F., Probl`emes de Cauchy et problèmes mixtes en thèorie des distributions, J. Analyse Math. Israel, 1959, vol. 7, pp. 105–187.

    Article  MATH  MathSciNet  Google Scholar 

  11. Lions, J.-L., Equations différentielles opérationnelles et problèmes aux limites, Berlin, 1961

    Book  MATH  Google Scholar 

  12. Yurchuk, N.I., Boundary Value Problems for Differential Equations with Operator Coefficients That Depend on a Parameter. II. Solvability and Properties of Solutions, Differ. Uravn., 1978, vol. 14, no. 5, pp. 859–870.

    MATH  MathSciNet  Google Scholar 

  13. Yurchuk, N.I., Boundary Value Problems for Differential Equations with Operator Coefficients That Depend on a Parameter. I. A Priori Estimates, Differ. Uravn., 1976, vol. 12, no. 9, pp. 1645–1661.

    MATH  Google Scholar 

  14. Korzyuk, V.I., The First Mixed Problem for a Second-Order Linear Hyperbolic Equation with Homogeneous Conditions in the Case of a Noncylindrical Domain, Differ. Uravn., 1992, vol. 28, no. 5, pp. 847–856.

    MATH  MathSciNet  Google Scholar 

  15. Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovom prostranstve (Linear Differential Equations in a Banach Space), Moscow: Nauka, 1967.

    Google Scholar 

  16. Berezanskii, Yu.M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov (Expansions in Eigenfunctions of Self-Adjoint Operators), Kiev: Naukova Dumka, 1965.

    MATH  Google Scholar 

  17. Lions, J.-L. and Magenes, E., Problemes aux limites non homogénes et applications, Paris: Dunod, 1968. Translated under the title Neodnorodnye granichnye zadachi i ikh prilozheniya, Moscow: Mir, 1971.

    MATH  Google Scholar 

  18. Lomovtsev, F.E., An Energy Inequality and a Method of GluingWeak Solutions of Hyperbolic Equations with Variable Domains of Piecewise Smooth Operators, Dokl. Akad. Nauk, 2013, vol. 448, no. 3, pp. 261–265.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    F. E. Lomovtsev

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  1. F. E. Lomovtsev
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Correspondence to F. E. Lomovtsev.

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Original Russian Text © F.E. Lomovtsev, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 5, pp. 646–657.

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Lomovtsev, F.E. Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators. Diff Equat 50, 643–654 (2014). https://doi.org/10.1134/S0012266114050073

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  • DOI: https://doi.org/10.1134/S0012266114050073

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