Advertisement

Ukrainian Mathematical Journal

, Volume 51, Issue 3, pp 366–376 | Cite as

On certain nonlinear pseudoparabolic variational inequalities without initial conditions

  • S. P. Lavrenyuk
  • M. B. Ptashnyk
Article
  • 16 Downloads

Abstract

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variablet. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior ast→−∞.

Keywords

Variational Inequality Heat Transfer Nonlinear Parabolic Equation Nonlocat Condition Double Porosity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. I. Barenblat, Yu. P. Zheltov, and I. N. Kochina, “On basic representations of the theory of filtration of homogeneous fluids in cracked rocks,”Prikl. Mat. Mekh.,24, Issue 3, 852–864 (1960).Google Scholar
  2. 2.
    L. I. Rubinshtein, “On the process of heat transfer in heterogeneous media,”Izv. Akad. Nauk SSSR. Ser. Geograf. Geofiz.,12, No. 1, 27–45 (1948).MathSciNetGoogle Scholar
  3. 3.
    A. F. Chudnovskii,Thermal Physics of Soils [in Russian], Nauka, Moscow (1976).Google Scholar
  4. 4.
    M. Majchrovski, “On inverse problems with nonlocal conditions for parabolic systems of partial differential equations and pseudoparabolic equations,”Demonstr. Math.,26, No. 1, 255–275 (1993).Google Scholar
  5. 5.
    T. W. Ting, “Parabolic and pseudoparabolic partial differential equations,”J. Math. Soc. Jpn.,21, No. 3, 440–453 (1969).zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    H. Gajewski, K. Gröger, and K. Zacharias,Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin (1974).zbMATHGoogle Scholar
  7. 7.
    V. R. Gopala Rao and T. W. Ting, “Initial-boundary value problems for pseudoparabolic partial differential equations,”Indiana Univ. Math. J.,23, No. 2, 131–153 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. E. Showalter, “Partial differential equations of Sobolev-Galpern type,”Pacif. J. Math.,31, No. 3, 787–793 (1969).zbMATHMathSciNetGoogle Scholar
  9. 9.
    W. Rundel, “The solution of initial-boundary value problem for pseudoparabolic partial differential equations,”Proc. Roy. Soc. Edinburgh,A 74 311–326 (1976).Google Scholar
  10. 10.
    D. Colton, “Pseudoparabolic equations in one space variable,”J. Different. Equat.,12, No. 3, 559–565 (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    I. V. Suveika, “Mixed problems for a nonstationary equation,”Mat. Issled., No. 58, 99–123 (1980).MathSciNetGoogle Scholar
  12. 12.
    M. Kh. Shkhanukov, “On some boundary-value problems for third-order equations arising on the modelling of fluid filtration in porous media,”Differents. Uravn.,18, No. 10, 689–699 (1982).zbMATHMathSciNetGoogle Scholar
  13. 13.
    J. P. Cannon, and Lin Jamping, “Classical and weak solution for one-dimensional pseudoparabolic equations with typical boundary data,”Ann Mat. Pure Appl., No. 152, 375–389 (1988).zbMATHCrossRefGoogle Scholar
  14. 14.
    M. O. Bas, and S. P. Lavrenyuk, “On the uniqueness of a solution of the Fourier problem for a system of Sobolev-Galpern type,”Ukr. Mat. Zh.,48, No. 1, 124–128 (1996).MathSciNetGoogle Scholar
  15. 15.
    M. O. Bas, and S. P. Lavrenyuk,The Fourier Problem for a Nonlinear Pseudoparabolic System [in Russian], Dep. at GPNTB of Ukraine No. 2017-Uk.95, Kiev (1995).Google Scholar
  16. 16.
    J.-L. Lions,Quelques Méthodes de Resolution des Problémes aux Limites Non Linéaires [Russian translation], Mir, Moscow (1972).zbMATHGoogle Scholar
  17. 17.
    A. A. Pankov,Bounded and Almost Periodic Solutions of Nonlinear Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  18. 18.
    S. P. Lavrenyuk, “Parabolic variational inequalities without initial conditions,”Differents. Uravn.,32, No. 10, 1–5 (1996).MathSciNetGoogle Scholar
  19. 19.
    S. P. Lavrenyuk and M. B. Ptashnyk, “Pseudoparabolic variational inequalities without initial conditions,” Ukr. Mat. Zh.50, No. 7, 919–929 (1998).CrossRefMathSciNetGoogle Scholar
  20. 20.
    N. M. Bokalo, “On the problem without initial conditions for some classes of nonlinear parabolic equations”,Tr. Sem. Petrovskogo, Issue 14, 3–44 (1989).zbMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • S. P. Lavrenyuk
  • M. B. Ptashnyk

There are no affiliations available

Personalised recommendations