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On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ3

In memory of Professor Vladimir Vragov (1945–2002)


Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.

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  1. 1.

    S. A. Aldashev, “A criterion for the existence of eigenfunctions of the Darboux-Protter spectral problem for degenerating multidimentional hyperbolic equations,” Differ. Equations 41(6), 833–839 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    A. Aziz and M. Schneider, “Frankl-Morawetz problems in R3,” SIAM J. Math. Anal. 10(5), 913–921 (1979).

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    J. Barros-Neto and I. Gelfand, “Fundamental solutions for the Tricomi operator,” Duke Math. J. 128(1), 119–140 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    A. V. Bitsadze, Some Classes of Partial Differential Equations (Gordon & Breach, New York, 1988).

    MATH  Google Scholar 

  5. 5.

    D. Edmunds and N. Popivanov, “A nonlocal regularization of some over-determined boundary-value problems,” SIAM J. Math. Anal. 29(1), 85–105 (1998).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    P. R. Garabedian, “Partial differential equations with more than two independent variables in the complex domain,” J. Math. Mech. 9(2), 241–271 (1960).

    MathSciNet  MATH  Google Scholar 

  7. 7.

    W. Haack and W. Wendland, Vorlesungen über Partielle und Pfaffsche Differentialgleichungen (Birkhäuser, Basel-Stuttgart, 1969) [in German].

    MATH  Google Scholar 

  8. 8.

    T. Hristov and N. Popivanov, “Singular solutions to Protter’s problem for a class of 3-D weakly hyperbolic equations,” Compt.Rend. Acad. Bulg. Sci. 60(7), 719–724 (2007).

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Khe Kan Cher, “On nontrivial solutions of some homogeneous boundary value problems for the multidimensional hyperbolic Euler-Poisson-Darboux equation in an unbounded domain,” Differ. Equations 34, 139–142 (1998).

    Google Scholar 

  10. 10.

    D. Lupo, C. Morawetz, and K. Payne, “On closed boundary value problems for equations of mixed elliptichyperbolic type,” Commun. Pure Appl. Math. 60(9), 1319–1348 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    D. Lupo, K. Payne, and N. Popivanov, “Nonexistence of nontrivial solutions for supercritical equations of mixed elliptic-hyperbolic type,” in Progress in Non-Linear Differential Equations and Their Applications 66 (Birkhäuser, Basel, 2006), pp. 371–390.

    Google Scholar 

  12. 12.

    T. E. Moiseev, “On the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions,” Differ. Equations 45(10), 1547–1549 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    C. Morawetz, “Mixed equations and transonic flow,” J. Hyperbolic Differ. Equations 1(1), 1–26 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    B. Opic and A. Kufner, Hardy-type Inequalities, in vol. 219 of Pitman Research Notes in Mathematics Series (Longman, Harlow, 1990).

    Google Scholar 

  15. 15.

    N. A. Orshubekov, “The Tricomi problem for some classes degenerating multidimensional mixed hyperbolic-parabolic type equations,” Proc. Adygea Internat. Academy of Sci. 11(2), 36–45 (2009).

    Google Scholar 

  16. 16.

    N. Popivanov, T. Popov, and R. Scherer, “Asymptotic expansions of singular solutions for 3 + 1 — D Protter problems,” J. Math. Anal. Appl. 331(2), 1093–1112 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    N. Popivanov and M. Schneider, “The Darboux problems in R 3 for a class of degenerating hyperbolic equations,” J. Math. Anal. Appl. 175(2), 537–578 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    M. Protter, “New boundary value problem for the wave equation and equations of mixed type,” J. Ration. Mech. Anal. 3, 435–446 (1954).

    MathSciNet  MATH  Google Scholar 

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Correspondence to T. D. Hristov.

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The text was submitted by the authors in English.

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Hristov, T.D., Popivanov, N.I. & Schneider, M. On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ3 . Sib. Adv. Math. 21, 262–273 (2011).

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  • mixed-type equation
  • boundary value problem
  • generalized solution
  • quasi-regular solution
  • uniqueness