Cauchy problem for equations with multiple characteristics

AbstractThis paper contains a proof of γn(χ) correctness of the noncharacteristic Cauchy problem for nonstrictly hyperbolic equations with analytic coefficients under the condition that its characteristic roots are smooth and under some additional assumptions on the lower-order terms. There are two extreme cases: (1) χ < rr − 1. In this case condition (0.6) is “void,” and we do not require conditions on Ps for s < m. For this case, see [3, 8]. (2) Case of constant multiplicity of characteristic roots and χ = +∞. In this case condition (0.6) implies conditions on Ps, where s = m, m − 1,…, m − r + 1, i.e., up to the same order as the necessary condition for C∞-correctness [2]. Recall that in the case of equations with characteristics of constant multiplicity condition (0.6) (Levi's condition in this case) for χ = ∞ is necessary [2, 4] and sufficient [1] for C∞-correctness