A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In our previous work, we showed that if the Hamilton map has a Jordan block of size $4$ on the double characteristic manifold $Σ$ of codimension $3$, then the Cauchy problem for $P$ is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and that this result is optimal. Moreover, if there are no bicharacterisitcs tangent to $Σ$, then the Cauchy problem is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and this result is also optimal. In the present paper, we remove the restriction on the codimension of $Σ$, thereby completing the result.

💡 Research Summary

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The paper investigates the Cauchy problem for second‑order hyperbolic differential operators (P) that possess non‑effectively hyperbolic double characteristics. Such operators are of the form
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