Advertisement

Ukrainian Mathematical Journal

, Volume 64, Issue 11, pp 1688–1697 | Cite as

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

  • I. I. Kovtun
  • M. N. Feller
Article
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 :

Keywords

Hilbert Space Wave Equation Cauchy Problem Orthonormal Basis English Translation 
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.
    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).Google Scholar
  2. 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).MathSciNetzbMATHGoogle Scholar
  3. 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).MathSciNetCrossRefGoogle Scholar
  4. 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).Google Scholar
  5. 5.
    P. Lévy, Problémes Concrets d’Analyse Fonctionnelle, Gauthier, Paris (1951).zbMATHGoogle Scholar
  6. 6.
    M. N. Feller, The Lévy Laplacian, Cambridge University Press, Cambridge (2005).zbMATHCrossRefGoogle Scholar
  7. 7.
    G. E. Shilov, “Some problems of analysis in Hilbert spaces. I,” Funkts. Anal. Prilozhen., 1, No. 2, 81–90 (1967).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • I. I. Kovtun
    • 1
  • M. N. Feller
    • 2
  1. 1.Ukrainian National University of Biological Resources and Nature ManagementKievUkraine
  2. 2.“Resource” Ukrainian Scientific-Research InstituteKievUkraine

Personalised recommendations