Abstract

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.

中文翻译：

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具有二次非线性积分微分方程的半轴问题

摘要

考虑一个函数方程，其中将两个变量函数及其时间导数的线性组合设置为等于相同函数关于空间变量的二次表达式的双积分。对于得到的具有二次非线性的积分微分方程，研究了初始数据连续且以正半轴为界的柯西问题。证明了经典逐次逼近方法的收敛性。根据迭代解的指数来估计近似的精度。证明了该问题在相关的函数空间中具有一个解决方案，并且建立了该解决方案的唯一性。从相关的适度类推导解决方案的先验估计。