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ON THE CAUCHY PROBLEM FOR NONEFFECTIVELY HYPERBOLIC OPERATORS: THE GEVREY 3 WELLPOSEDNESS
No full textFor hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa
On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness
No full textFor a hyperbolic second-order differential operator , 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 . If the Hamilton map of (the linearization of the Hamilton field 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 is well posed in any Gevrey class for any lower-order term. In this paper we prove that if is noneffectively hyperbolic and, moreover, such that on a double characteristic manifold of codimension , assuming that there is no null bicharacteristic landing tangentially, then the Cauchy problem for is well posed in the Gevrey class for any lower-order term (strong Gevrey well-posedness with threshold ), extending in particular via energy estimates a previous result of Hörmander in a model case
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