The Cauchy problem for hyperbolic operators of strong type
Let \(P(x,D) = P_m (x,D) + P_{m - 1} (x,D) + \cdots + P_0(x)\) be a differential operator of order \(m\) in the variables \(x = (x_1, \ldots, x_n)\) (here \(x_1\) can be interpreted as a time variable), \(P_j\) being its homogeneous part of order \(j\). The authors are concerned with the \(C^\infty\) well posedness of the Cauchy problem for \(P\) with respect to the hyperplanes \(x_1 = \text{const}\) in the case of multiple characteristics. For sake of simplicity we shall assume that the multiplicity of the characteristic roots of \(P_m\) is \(m\). The celebrated Mizohata and Ivrii-Petkov theorems tell us that, for arbitrary \(m\), if the Cauchy problem for \(P\) is well posed, then \(P_m\) is hyperbolic and, if \(P_m\) vanishes at \(\rho\) of order \(m\), then \(P_j\) vanishes at \(\rho\) of order \([m - 2j]_+\). Moreover it is known that in general these conditions are not enough to ensure the well posedness of the Cauchy problem for \(P\). When they are, we say that \(P\) is strongly hyperbolic. The authors of the present paper prove that a certain class of hyperbolic operators with characteristics of higher multiplicity are strongly hyperbolic. It is too long to give details on the assumptions here, but one can roughly describe the class considered by the authors saying that the difference of any two characteristic roots of the operator is bounded from below by a time function vanishing on the characteristic manifold, or, equivalently, that the propagation cone of (a localization of) \(P\) at a multiple point is transverse to the multiple characteristic variety. The method of proof, which contains also a propagation of singularities result, consists in proving first a microlocal a priori estimate in spaces with a pseudodifferential weight function. It is worth noting that the class considered by the authors falls presumably very short of the class of all strongly hyperbolic operators; a discussion of this issue is given in the introductory section of the paper.
\(C^ \infty\) well posedness of the Cauchy problem
scientific article; zbMATH DE number 706692
scientific article