Skip to main content
Log in

Boundary-value problems for a nonlinear hyperbolic equation with Lévy Laplacian

  • Published:
Ukrainian Mathematical Journal Aims and scope

We present solutions of the boundary-value problem

$$ U\left( {0,x} \right)={u_0},\,\,\,\,U\left( {t,0} \right)={u_1} $$

and the external boundary-value problem

$$ U\left( {0,x} \right)={v_0},\,\,\,\,\,U\left( {t,x} \right){|_{\varGamma }}={v_1},\,\,\,\,\mathop{\lim}\limits_{{\left\| x \right\|H\to \infty }}U\left( {t,x} \right)={v_2} $$

for the nonlinear hyperbolic equation

$$ \frac{{{\partial^2}U\left( {t,x} \right)}}{{\partial {t^2}}}+\alpha \left( {U\left( {t,x} \right)} \right){{\left[ {\frac{{\partial U\left( {t,x} \right)}}{{\partial t}}} \right]}^2}={\varDelta_L}U\left( {t,x} \right) $$

with infinite-dimensional Lévy Laplacian Δ L :

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

Access this article

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. M. N. Feller, “Boundary-value problems for the wave equation with Lévy Laplacian in the Gâteaux class,” Ukr. Mat. Zh., 61, No. 11, 1564–1574 (2009); English translation: Ukr. Math. J., 61, No. 11, 1839–1852 (2009).

  2. S. Albeverio, Ya. I. Belopolskaya, and M. N. Feller, “Boundary problems for the wave equation with the Lévy Laplacian in Shilov’s class,” Meth. Funct. Anal. Topol., 16, No. 3, 197–202 (2010).

    MathSciNet  MATH  Google Scholar 

  3. S. A. Albeverio, Ya. I. Belopolskaya, and M. N. Feller, “Cauchy problem for the wave equation with Lévy Laplacian,” Mat. Zametki, 87, Issue 6, 803–813 (2010).

    Article  MathSciNet  Google Scholar 

  4. M. N. Feller, “Boundary-value problems for a nonlinear hyperbolic equation with divergent part and Lévy Laplacian,” Ukr. Mat. Zh., 64, No. 2, 237–244 (2012); English translation: Ukr. Math. J., 64, No. 2, 273–281 (2012).

  5. P. Lévy, Problémes Concrets d’Analyse Fonctionnelle, Gauthier, Paris (1951).

    MATH  Google Scholar 

  6. M. N. Feller, The Lévy Laplacian, Cambridge University Press, Cambridge (2005).

    Book  MATH  Google Scholar 

  7. G. E. Shilov, “Some problems of analysis in Hilbert spaces. I,” Funkts. Anal. Prilozhen., 1, No. 2, 81–90 (1967).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 11, pp. 1492–1499, November, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovtun, I.I., Feller, M.N. Boundary-value problems for a nonlinear hyperbolic equation with Lévy Laplacian. Ukr Math J 64, 1688–1697 (2013). https://doi.org/10.1007/s11253-013-0744-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0744-3

Keywords

Navigation