Siberian Advances in Mathematics

, Volume 29, Issue 2, pp 128–136 | Cite as

Convergence of the Successive Approximation Method in the Cauchy Problem for an Integro-Differential Equation with Quadratic Nonlinearity

  • V. L. VaskevichEmail author
  • A. I. ShcherbakovEmail author


The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on any finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.


nonlinear integro-differential equation quadratic nonlinearity Cauchy problem existence theorem successive approximation method a priori estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. M. Vainberg, “Integro-differential equations,” Itogi Nauki, Mat. Anal. Teor. Veroyatn. Regul. 1962, 5–37 (VINITI, Moscow, 1964).Google Scholar
  2. 2.
    M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations (Nauka, Moscow, 1969; Noordhoff International Publishing, Leyden, 1974).zbMATHGoogle Scholar
  3. 3.
    N. K. Bell, V. N. Grebenev, S. B. Medvedev, and S. V. Nazarenko, “Self-similar evolution of Alfven wave turbulence,” J. Phys. A: Math. Theor. 50, No. 43, p. 14 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. Galtier, “Wave turbulence in astrophysics,” Advances in Wave Turbulence. World Scientific Series on Nonlinear Science, Ser. A. 83,p. 73 (World Scientific Publishing, Singapore, 2013).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Galtier, S. Nazarenko, A. C. Newell, and A. Pouquet, “A weak turbulence theory for incompressible magnetohydrodynamics,” J. Plasma Physics 63, 447 (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    S. Galtier and E. Buchlin, “Nonlinear diffusion equations for anisotropic magnetohydrodynamic turbulence with cross-helicity,” The Astrophysical J. 722, 1977 (2010).CrossRefGoogle Scholar
  7. 7.
    S. Nazarenko, Wave Turbulence. Lecture Notes in Physics, No. 825 (Springer, Berlin, Heidelberg, 2011).CrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations