Analysis of micro local structure of hyperbolic equations and characterization of hyperbolic equations for which the Cauchy problem is well-posed

我考虑了二阶双曲型算子的柯西问题，其主部的系数仅取决于时间变量。在系数是时间变量的真实的解析函数的情况下，我得到了C^∞适定性的一个充分条件。证明了当空间维数小于3或主部系数为半代数函数时，这个充分条件也是必要条件。例如，在一个实施例中，多项式）的时间变量。在此充分条件下，证明了Cauchy问题解的奇性是沿着破零双特征线传播的.

I considered the Cauchy problem for second-order hyperbolic operators with the coefficients of their principal parts depending only on the time variable. In the case where the coefficients are real analytic functions of the time variable, I obtained a sufficient condition for C^∞ well-posedness. And I showed that this sufficient condition is also a necessary one when the space dimension is less than 3 or the coefficients of the principal part are semi-algebraic functions (e. g., polynomials) of the time variable. Moreover, under this sufficient condition I proved that the singularities of solutions to the Cauchy problem propagate along broken null bicharacteristics.