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

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.