Advertisement

Global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\)

  • Cung The Anh
Article

Abstract

The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\) with the locally Lipschitz nonlinearity satisfying a subcritical growth condition.

Mathematics Subject Classification (2010)

35D35 35K65 35B41 

Keywords

Strongly degenerate Unbounded domains Mild solution Global attractor Tail-estimates method Sectorial operator Lyapunov function 

References

  1. 1.
    Anh C.T., Hung P.Q., Ke T.D., Phong T.T.: Global attractor for a semilinear parabolic equation involving Grushin operator. Electron. J. Differ. Equ. 32, 1–11 (2008)MathSciNetGoogle Scholar
  2. 2.
    Anh C.T., Ke T.D.: Existence and continuity of global attractor for a semilinear degenerate parabolic equation. Electron. J. Differ. Equ. 61, 1–13 (2009)MathSciNetGoogle Scholar
  3. 3.
    Anh C.T., Tuyet L.T.: Strong solutions to a strongly degenerate semilinear parabolic equation. Vietnam J. Math. 41, 217–232 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anh C.T., Tuyet L.T.: On a semilinear strongly degenerate parabolic equation in unbounded domains. J. Math. Sci. Univ. Tokyo 20, 91–113 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chepyzhov V.V., Vishik M.I.: Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publication, vol. 49. Amererican Mathematical Society, Providence (2002)Google Scholar
  6. 6.
    Franchi B., Lanconelli E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 523–541 (1983)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Conference on linear partial and pseudodifferential operators (Torino, 1982). Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue 105–114 (1984)Google Scholar
  8. 8.
    Franchi B., Lanconelli E.: An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Commun. Partial Differ. Equ. 9, 1237–1264 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grushin V.V.: A certain class of elliptic pseudo differential operators that are degenerated on a submanifold. Mat. Sb. 84(126), 163–195 (1971)MathSciNetGoogle Scholar
  10. 10.
    Han Y., Zhao Q.: A class of Caffarelli–Kohn–Nirenberg type inequalities for generalized Baouendi–Grushin vector fields. Acta Math. Sci. Ser. A Chin. Ed. 31, 1181–1189 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Henry D.: Geometric theory of semilinear parabolic equations. Lecture notes in mathematics. Springer, Berlin (1981)Google Scholar
  12. 12.
    Kogoj A.E., Lanconelli E.: On semilinear Δλ-Laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kogoj A., Sonner S.: Attractors for a class of semi-linear degenerate parabolic equations. J. Evol. Equ. 13, 675–691 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lanconelli E., Kogoj A.E.: X-elliptic operators and X-control distances. Ric. Mat. 49(Suppl.), 223–243 (2000)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Markasheva V.A., Tedeev A.F.: Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi–Grushin type. Math. Notes 85, 385–396 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Markasheva V.A., Tedeev A.F.: The Cauchy problem for a quasilinear parabolic equation with gradient absorption. Mat. Sb. 203, 131–160 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mihilescu M., Moroanu G., Stancu-Dumitru D.: Equations involving a variable exponent Grushin-type operator. Nonlinearity 24, 2663–2680 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Prizzi M.: A remark on reaction-diffusion equations on unbounded domains. Discret. Cont. Dyn. Syst. 9, 281–286 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Prizzi M., Rybakowski K.: Attractors for reaction-diffusion equations on arbitrary unbounded domains. Topol. Methods Nonlinear Anal. 30, 251–277 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Robinson J.C.: Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge (2001)Google Scholar
  21. 21.
    Thuy P.T., Tri N.M.: Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. NoDEA 19, 279–298 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Thuy P.T., Tri N.M.: Long-time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators. NoDEA 20, 1213–1224 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Temam R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang B.: Attractors for reaction-diffusion equations in unbounded domains. Physica D 179, 41–52 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationCau Giay, HanoiVietnam

Personalised recommendations