Necessary conditions for well-posedness of the flat Cauchy problem and the regularity-loss of solutions (original) (raw)

1983, Publications of the Research Institute for Mathematical Sciences

It is well-known that for some weakly hyperbolic operators, the regularityloss of solutions depends largely on lower order terms. For example, if P has the principal part 3|-t 2k d 2 x and the Cauchy problem for P with the initial surface £=0 is well-posed, then the lower order terms have the form ad t + t k~1 bd x +c where a, b, c are C°°-functions. And the regularity-loss depends on \Reb(Q, x)\. In this paper, we pay attention to the behavior of the principal part with respect to the time variable t, and we give a result of the following type (cf. [2], [3] and their references). Assume that the (flat) Cauchy problem is well-posed. Then, the lower order terms satisfy some conditions, and the regularity-loss of solutions depends on certain quantities, which are determined by lower order terms. We observe another example. For the operator P=d z-td z x +ad t +bd x +c where a, b, c are C°°-functions, the regularity-loss of solutions does not depend on lower order terms. But, if we consider the operator P=d z-td z x +ad t +t~1 /z bd x +c, then the regularity-loss depends on |Re6(0, x)\. We want also to deal Fuchsian operators. So, we consider the operators whose coefficients may have fractional or negative powers of t. For these operators, we can consider the flat Cauchy problem to which the non-characteristic Cauchy problem for operators with C°°-coefficients can be easily reduced. Our program is as follows. In Section 1, we state definitions, the result and some examples. Our result consists of three theorems. In Section 2, we consider two transformations of operators which reduce the theorems to easier situation. In Section 3, we study an elementary fact on Newton polygons and apply it. In Sections 4, 5 and 6, we prove the theorems.