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On the Cauchy problem for noneffectively hyperbolic operators: the Gevrey 4 well-posedness. (English) Zbl 1276.35112

Summary: For a hyperbolic second-order differential operator \(P\), we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol \(p\). If the Hamilton map \(F_{p}\) of \(p\) (the linearization of the Hamilton field \(H_{p}\) along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for \(P\) is well posed in any Gevrey class \(1\leq s < +\infty\) for any lower-order term. In this paper we prove that if \(p\) is noneffectively hyperbolic and, moreover, such that \(\text{Ker}F_{p}^{2}\cap \text{Im}F_{p}^{2}\neq\{0\}\) on a \(C^{\infty}\) double characteristic manifold \(\Sigma\) of codimension 3, assuming that there is no null bicharacteristic landing \(\Sigma\) tangentially, then the Cauchy problem for \(P\) is well posed in the Gevrey class \(1\leq s < 4\) for any lower-order term (strong Gevrey well-posedness with threshold 4), extending in particular via energy estimates a previous result of Hörmander in a model case.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35C10 Series solutions to PDEs
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