We study a class of third order hyperbolic operators $P$ in $G = \{(t, x):0\leq t \leq T, x \in U \Subset {\mathbb R}^{n}\}$ with triple characteristicsat $\rho = (0, x_0, \xi), \xi \in {\mathbb R}^n \setminus \{0\}$. We considerthe case when the fundamental matrix of the principal symbol of $P$ at $\rho$has a couple of non-vanishing real eigenvalues. Such operators are called {\iteffectively hyperbolic}. V. Ivrii introduced the conjecture that everyeffectively hyperbolic operator is {\it strongly hyperbolic}, that is theCauchy problem for $P + Q$ is locally well posed for any lower order terms $Q$.This conjecture has been solved for operators having at most doublecharacteristics and for operators with triple characteristics in the case whenthe principal symbol admits a factorization. A strongly hyperbolic operator in$G$ could have triple characteristics in $G$ only for $t = 0$ or for $t = T$.We prove that the operators in our class are strongly hyperbolic if $T$ issmall enough. Our proof is based on energy estimates with a loss of regularity.
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机译：我们研究$ G = \ {（t，x）：0 \ leq t \ leq T，x \ in U \ subset {\ mathbb R} ^ {n} \} $中的一类三阶双曲算子$ P $在$ \ rho =（0，x_0，\ xi），\ xi \ in {\ mathbb R} ^ n \ setminus \ {0 \} $中具有三重特征。我们考虑以下情况：$ \ rho $的$ P $的主要符号的基本矩阵具有几个不变的真实特征值。这样的运算符称为{\ itfully双曲线}。 V. Ivrii提出了一个猜想，即每个有效双曲算子都是{\它强烈双曲}，即对于$ P + Q $的Cauchy问题在任何低阶项$ Q $上都是适当的。该猜想已为具有在主符号允许分解的情况下，大多数双特征和具有三重特征的运算符。 $ G $中的强双曲算子仅在$ t = 0 $或$ t = T $时可以在$ G $中具有三重特征。我们证明，如果$ T $足够小，则此类中的算子是强双曲的。我们的证明是基于不规则的能量估计。
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