On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness

Abstract

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 Fp of p (the linearization of the Hamilton field Hp 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≤s<+∞ for any lower-order term. In this paper we prove that if p is noneffectively hyperbolic and, moreover, such that KerFp2∩ImFp2≠{0} on a C∞ double characteristic manifold Σ of codimension 3, assuming that there is no null bicharacteristic landing Σ tangentially, then the Cauchy problem for P is well posed in the Gevrey class 1≤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.