Cauchy Problem

one of the fundamental problems of the theory of differential equations, first studied systematically by A. Cauchy. It consists in finding a solution u(x, t), for x = (x_i, …, x_n), of a differential equation of the form

satisfying the initial conditions

where G₀—the carrier of the initial data—is a region in the hyperplane t = t₀ of the space of variables x₁, …, x_n. When F and f_k, for k = 0, …, m — 1, are analytic functions of their arguments, then the Cauchy problem (1), (2) always has a unique solution in some region G of the space of variables t, x containing G₀. This solution, however, can prove to be unstable (that is, a small change in the initial data can cause a large change in the solution), for example, for cases when equation (1) is elliptic. If equation (1) is not hyperbolic and the initial data are not analytic, then the Cauchy problem (1), (2) can lose meaning.