Abstract
In this paper, we prove that for non-effectively hyperbolic operators with smooth double characteristics exhibiting a Jordan block of size 4 on the double manifold, the Cauchy problem is well-posed in the Gevrey 5 class, beyond the generic Gevrey class 2 (see, e.g., [5]). Moreover, we show that this value is optimal, due to certain geometric constraints on the Hamiltonian flow of the principal symbol. These results, together with results already proved, give a complete picture of the well-posedness of the Cauchy problem around hyperbolic double characteristics.
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Bernardi, E., Nishitani, T. On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness. J Anal Math 105, 197–240 (2008). https://doi.org/10.1007/s11854-008-0035-3
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DOI: https://doi.org/10.1007/s11854-008-0035-3