Advertisement

Ukrainian Mathematical Journal

, Volume 52, Issue 5, pp 789–802 | Cite as

Notes on infinite-dimensional nonlinear parabolic equations

  • M. N. Feller
Article

Abstract

We present a method for the solution of the Cauchy problem for three broad classes of nonlinear parabolic equations
$$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {\Delta _L U\left( {t,x} \right)} \right), \frac{{\partial U\left( {t,x} \right)}}{{\partial t}} f\left( {t,\Delta _L U\left( {t,x} \right)} \right),$$
and
$$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {U\left( {t,x} \right), \Delta _L U\left( {t,x} \right)} \right)$$
with the infinite-dimensional Laplacian ΔL.

Keywords

Cauchy Problem Nonlinear Equation Dirichlet Problem Differentiable Function Form Versus 
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.
    P. Levi, Problémes Concrets d’Analyse Fonctionnelle, Gauthier-Villars, Paris 1951.Google Scholar
  2. 2.
    L. Accardi, P. Gibilisco, and I. V. Volovich, The Lévy Laplacian and the Yang-Mills equations, Preprint No. 129, Vito Volterra Centre, Rome (1992), pp. 1–7.Google Scholar
  3. 3.
    G. E. Shilov, “On some problems of analysis in Hilbert spaces. III,” Mat. Sb., 74, No. 1, 161–168 (1967).MathSciNetGoogle Scholar
  4. 4.
    M. N. Feller, “On one nonlinear equation unsolved with respect to the Levy Laplacian,” Ukr. Mat. Zh., 48, No. 5, 719–721 (1996).CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. N. Feller, “The Riquier problem for a nonlinear equation resolved with respect to the iterated Levy Laplacian,” Ukr. Mat. Zh., 50, No. 11, 1574–1577(1998).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. N. Feller, “The Riquier problem for a nonlinear equation unresolved with respect to the iterated Levy Laplacian,” Ukr. Mat. Zh., 51. No. 3, 423–427 (1999).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. B. Sokolovskii, “Infinite-dimensional parabolic equations with Levy Laplacians and certain variation problems.” Sib. Mat. Zh., 35. No. 1, 177–180 (1994).MathSciNetGoogle Scholar
  8. 8.
    M. N. Feller, “Infinite-dimensional elliptic equations and operators of the Levy type,” Usp. Mat. Nauk, 41, No. 4, 97–140 (1986).MathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • M. N. Feller
    • 1
  1. 1.UkrNIMODKiev

Personalised recommendations