TY - JOUR

T1 - Problems on a Semiaxis for an Integro-Differential Equation with Quadratic Nonlinearity

AU - Vaskevich, V. L.

N1 - Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - A functional equation is considered in which a linear combination of a two-variable function and its time derivative is set equal to the double integral of a quadratic expression of the same function with respect to space variables. For the resulting integro-differential equation with quadratic nonlinearity, the Cauchy problem with initial data continuous and bounded on the positive semiaxis is investigated. The convergence of the classical method of successive approximations is proved. The accuracy of the approximation is estimated depending on the index of the iterative solution. It is proved that the problem has a solution in associated function spaces, and the uniqueness of this solution is established. An a priori estimate for solutions from the associated well-posedness class is derived. A guaranteed time interval of solution existence is found.

AB - A functional equation is considered in which a linear combination of a two-variable function and its time derivative is set equal to the double integral of a quadratic expression of the same function with respect to space variables. For the resulting integro-differential equation with quadratic nonlinearity, the Cauchy problem with initial data continuous and bounded on the positive semiaxis is investigated. The convergence of the classical method of successive approximations is proved. The accuracy of the approximation is estimated depending on the index of the iterative solution. It is proved that the problem has a solution in associated function spaces, and the uniqueness of this solution is established. An a priori estimate for solutions from the associated well-posedness class is derived. A guaranteed time interval of solution existence is found.

KW - a priori estimate

KW - Cauchy problem

KW - existence theorem

KW - integro-differential equation

KW - quadratic nonlinearity

KW - successive approximations

UR - http://www.scopus.com/inward/record.url?scp=85086049603&partnerID=8YFLogxK

U2 - 10.1134/S0965542520040181

DO - 10.1134/S0965542520040181

M3 - Article

AN - SCOPUS:85086049603

VL - 60

SP - 590

EP - 600

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 4

ER -