On the Cauchy Problem for Weakly Hyperbolic Systems

Abstract

where the coefficients A](x,f) and B(x,i) are (in, in) matrices whose elements belong to the class J3 (fl) (in the sense of L. Schwartz [8]). As is well-known, for the <£-well-posedness of (1 • 1) the characteristic roots Ai(:r, t <?), (z = l, 2, • • • , m) (the roots of det (tl—^^A^x, t) $i)=0) must be real at any point (x, t Q^QxRg (S. Mizohata [4], P. D. Lax [2]). Moreover, under the assumption that the roots h(x, t\£) have constant multiplicities, the matrix A(x, t', $) =^2j=lAj(x, t^)gj must be diagonalizable at any poit (x9t\f) for the problem (1-1) to be c^-well-posed for every lower order term B(x,£) (K. Kajitani [1]). Now we consider the case that A(x,t\tT) is not diagonalizable. To discuss such a case V. M. Petkov has used the method of asymptotic expansions ([9], [10]). Quite different from his, our approach to this problem is much due to the so-called energy estimates (see S. Mizohata [4], S. Mizohata and Y. Ohya [6], [7]). The results of our present article can briefly be stated as follows.