The present paper is a survey of results on the correctness1 of the Cauchy
problem for operators both with constant and with variable coefficients. We
mainly concentrate on results that apply to as general operators as possible, in
particular, without distinguishing the principal part. Statements that apply for
specific classes of operators, such as hyperbolic, parabolic, etc., are discussed only
insofar as they illustrate general constructions. Such specific statements may be
found in more detail in papers devoted to the particular type of equation. In our
selection of results dealing with equations with constant coefficients, we chose
the ones at least a significant part of which also covers the case of variable
coefficients.
The Cauchy problem is a classical problem in partial differential equations: it
arises naturally in physical problems. The first examples of partial differential
equations appeared in the middle of the XVIIIth century in the framework of
mathematical physics. D'Alembert, Euler, and Bernoulli studied the equation of
a vibrating string from different points of view. The main goal, as they saw it,
was a derivation of a general solution (an integral) of the equation. The primary
observation was that while the general solution of an ordinary differential
equation depends on some arbitrary constants, a general solution of a partial differential
equation should depend on some arbitrary functions. D'Alembert's solution of
the string equation contained two arbitrary functions of one (space) variable. The
works of the classics of the XVIIIth century contained in an inchoate form many
basic ideas of the future theory: characteristics, separation of variables, expansion
of a solution in a basis (harmonics).