Bicharacteristic curves and wellposedness for hyperbolic equations with non-involutive multiple characteristics (Q1332794)

We consider a third order hyperbolic operator as \(P(x,D) = P_3 + P_2 + P_1 + P_0\). We assume the principal part \(P_3\) is the product of three first order operators \(a_i\), where the symbols of \(a_i\) are given by \(a_i = (\xi_0 - \Lambda_i)\) with \(\Lambda_i (x, \xi')\) real valued pseudodifferential operators of order 1 in \(x'\). We also asume the characteristics of \(a_i\) cross each other non-involutively, that is, \(\{a_i, a_j\} \neq 0\), for \(i \neq j\) on \(a_i = a_j\). Then we can divide the characteristics of the hyperbolic operator \(P\) into two types. We indicate the types by \(\text{sign} (a_0, a_1, a_2)\) which is defined as \(\text{sign} (a_0, a_1, a_2) = + 1\) if \(\{a_0, a_1\}\), \(\{a_1, a_2\}\) have the same sign \(+\) or \(-\), and \(\text{sign} (a_0, a_1, a_2) = - 1\) otherwise. The aim of the paper is to describe the consequences of the well posedness of the Cauchy problem for the type of the characteristics.