Skip to main content
SpringerLink
Log in

Nonexistence of Solutions of Some Non-Linear Non-Local Evolution Systems on the Heisenberg Group

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

We present non-existence results for systems of non-local in space hyperbolic equations, for systems of non-local in space parabolic equations, and for systems of non-local in space hyperbolic equations with linear damping terms. Our method of proof is based on the test function method with a help of a convexity inequality recently proved in [2].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Abe, Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion. Physica A 356, No 2–4 (2006), 403–407.

    MathSciNet  Google Scholar 

  2. A. Alsaedi, B. Ahmad, M. Kirane, Blowing-up solutions of some nonlinear non-local evolution equations on the Heisenberg group. Electron. J. Diff. Equ. 2015 (2015), Paper No 227, 1–10.

    Article  Google Scholar 

  3. H. Bahouri, I. Gallagher, Paraproduit sur le groupe de Heisenber et applications. Revista Mathematica Iberoamericana 17 (2001), 69–105.

    Article  Google Scholar 

  4. C.-H. Chang, Der-C.E. Chang, P. Greiner, H.-P. Lee, The positivity of the heat kernel on Heisenberg group. Anal. Appl. 11 (2013), # 1350019.

  5. A. Cordoba, D. Cordoba, A maximum principle applied to quasigeostrophic equations.Comunn. Math. Phys. 249 (2004), 511–528.

    Article  Google Scholar 

  6. A. Elhamidi, M. Kirane, Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group. Abstract and Applied Analysis 2004, No 2 (2004), 155–164.

    Article  MathSciNet  Google Scholar 

  7. H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut = Δu + up+1. J. Fac. Sci. Univ. Tokyo Sec. I 13 (1966), 109–124.

    Google Scholar 

  8. J.A. Goldstein, I. Kombe, Nonlinear degenerate parabolic equations on the Heisenberg group. Int. J. Evol. Equ. 1, No 1 (2005), 1–22.

    MathSciNet  MATH  Google Scholar 

  9. F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), 235–268.

    Article  MathSciNet  Google Scholar 

  10. A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80 (2009), # 022103.

  11. N. Ju, The maximum principle and the global attractor for the dissipative 2-D quasi-geostrophic equations. Comm. Pure Appl. Ana. (2005), 161–181.

    Google Scholar 

  12. T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations in three space dimensions. Comm. Pure Appl. Math. 37 (1984), 443–455.

    Article  MathSciNet  Google Scholar 

  13. N. Laskin, Fractional Schrödinger equation. Physical Review E 66 (2002), # 056108, 7 pages; available online: http://arxiv.org/abs/quant-ph/0206098.

  14. N. Laskin, Fractional quantum mechanics. Physical Review E 62 (2000), 3135–3145; available online: http://arxiv.org/abs/0811.1769.

    Article  Google Scholar 

  15. R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008).

    Book  Google Scholar 

  16. M. Kirane, M. Qafsaoui, Fujita’s exponent for a semilinear wave equation with linear damping. Advanced Nonlinear Studies 2, No 1 (2002), 41–49.

    Article  MathSciNet  Google Scholar 

  17. F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos, Solitons, Fractals 7 (1986), 1461–1477.

    Article  Google Scholar 

  18. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339 (2000), 1–77.

    Article  MathSciNet  Google Scholar 

  19. E. Mitidieri, S.I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Proc. of the Steklov Institute of Mathematics 234 (2001), 1–383.

    MATH  Google Scholar 

  20. D. Müller, E. M. Stein, Litp-estimates for the wave equation on the Heisenberg group. Rev. Mat. Iberoamericana 15, No 2 (1999), 297–334.

    Article  MathSciNet  Google Scholar 

  21. A.I. Nachman, The wave equation on the Heisenberg group. Comm. Partial Differential Equations 7, No 6 (1982), 675–714.

    Article  MathSciNet  Google Scholar 

  22. A. Pascucci, Semilinear equations on nilpotent Lie groups: global existence and blow-up of solutions. Matematiche 53 (1998), 345–357.

    MathSciNet  MATH  Google Scholar 

  23. M. Rammaha, Nonlinear wave equation in high dimensions. In: Differential Equations and Applications, Ohio Univ. Press (1989), 322–326.

    Google Scholar 

  24. T. Sideris, Nonexitence of global solution to semilinear wave equatins in high dimensions. J. Differential Equations 52 (1984), 378–406.

    Article  MathSciNet  Google Scholar 

  25. W.A. Strauss, Nonlinear Wave Equations. C.B.M.S Lecture Notes 73, Amer. Math. Soc., Providence, RI (1989).

    Google Scholar 

  26. V.E. Tarasov, Fractional Dynamics. Ser. Nonlinear Physical Science, Springer (2010).

    Book  Google Scholar 

  27. V.E. Tarasov, Fractional Heisenberg equation. Phys. Lett. A 372 (2008), 2984–2988.

    Article  MathSciNet  Google Scholar 

  28. G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping. J. Differential Equations 174 (2001), 464–489.

    Article  MathSciNet  Google Scholar 

  29. L. Véron, S.I. Pohozaev, Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group. Manuscripta Math. 102 (2000), 85–99.

    Article  MathSciNet  Google Scholar 

  30. Q.S. Zhang, The critical exponent of a reaction diffusion equation on some Lie groups. Math. Z. 228 (1998), 51–72.

    Article  MathSciNet  Google Scholar 

  31. Q.S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J. 97 (1999), 515–539.

    Article  MathSciNet  Google Scholar 

  32. Q.S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci. Paris S´erie I Math. 333 (2001), 109–114.

    Article  MathSciNet  Google Scholar 

  33. C. Zuily, Existence globale de solutions réguliéres pour l’équation des ondes non linéaires amortie sur le groupe de Heisenberg (In French) [Global existence of regular solutions for the damped nonlinear wave equation on the Heisenberg group]. Indiana Uni. Math. J. 62, No 2 (1993), 323–360.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Arts and Sciences, Dept. of Mathematics, P.O. Box 2713, Doha, Qatar

    Mokhtar Kirane

  2. Pôle Sciences et Technologies Avenue Michel Céepeau, Université de La Rochelle Laboratoire de Matématiques MIA, 17042, La Rochelle Cedex 1, France

    Mokhtar Kirane

Authors
  1. Mokhtar Kirane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mokhtar Kirane.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kirane, M. Nonexistence of Solutions of Some Non-Linear Non-Local Evolution Systems on the Heisenberg Group. FCAA 18, 1336–1349 (2015). https://doi.org/10.1515/fca-2015-0077

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2015-0077

Keywords

Search

Navigation