有效hyperbolic operators具有变重数三特征的Cauchy问题
Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity
摘要：
研究了一类在$G = \{(t, x):0 \leq t \leq T, x \in U \Subset {\mathbb R}^{n}\}$中的三阶双曲型算子$P$在$ρ= (0, x_0, ξ), ξ\in {\mathbb R}^n \setminus \{0\}$上具有三重特征.我们考虑当$P$在$ρ$的主符号的基本矩阵具有一对非零实特征值时的情况。这种算子称为{ \it有效双曲}。Ivrii提出了有效双曲算子是{ \it强双曲}的猜想，即$P + Q$的Cauchy问题对任意低阶项$Q$都是局部适定的.这个猜想已经解决了运营商最多有双重的特点和运营商的三重特点的情况下，当主符号承认一个因式分解。$G$中的强双曲算子仅对$t = 0$或对$t = T$有三重特征.证明了当$T$足够小时，我们类中的算子是强双曲的。我们的证明是基于能量估计与损失的规律性。
英文摘要：
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 characteristics at $ρ= (0, x_0, ξ), ξ\in {\mathbb R}^n \setminus \{0\}$. We consider the case when the fundamental matrix of the principal symbol of $P$ at $ρ$ has a couple of non-vanishing real eigenvalues. Such operators are called {\it effectively hyperbolic}. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is {\it strongly hyperbolic}, that is the Cauchy problem for $P + Q$ is locally well posed for any lower order terms $Q$. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the 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 $T$ is small enough. Our proof is based on energy estimates with a loss of regularity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
