101,269 research outputs found
On second order weakly hyperbolic equations and the Gevrey classes
No full textWe study the Cauchy problem for a second order weakly
hyperbolic operator with coefficients depending only on time. We
consider the case of coefficients of the principal part belonging
to an intermediate class between and the real analytic class
and we specify the function spaces in which the Cauchy problem
is well posed. Moreover we show by a counter example that this
results are in some sense optimal
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
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
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