论文标题
一类严格双曲线的库奇问题的全球良好性与系数不持续不变
Global Well-Posedness of a Class of Strictly Hyperbolic Cauchy Problems with Coefficients Non-Absolutely Continuous in Time
论文作者
论文摘要
我们研究了相对于相位空间上的一类指标，研究了在$ [0，t] \ times \ r^n $上定义的某些严格双曲线方程的解决方案的行为。特别是，我们研究了方程式的全球规律性和衰减问题，该方程式具有$ x $的多种系数，并以$ \ t $ dervivivient的订单$ \ textnormal {o}（t^{ - q}），$ q \ in \ big big [1，\ frac {3} {3} {2} $）。为此，定义了基于公制的合适的广义符号类，并使用关联的Planck函数来定义无限订单操作员以执行共轭。我们证明，该解决方案不仅经历了规律性的丧失（通常是针对以$ x $为界的系数的情况），而且还与针对广义符号类别的Sobolev空间定义的初始基准相关的衰减。此外，我们观察到，可以通过与给定方程系数的最佳选择来获得解决方案的精确行为。我们还在全球环境中得出锥条件。
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $[0,T]\times \R^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ and with their $t$-derivative of order $\textnormal{O}(t^{-q}),$ where $q \in \big[1,\frac{3}{2}\big)$. For this purpose, an appropriate generalized symbol class based on the metric is defined and the associated Planck function is used to define an infinite order operator to perform conjugation. We demonstrate that the solution not only experiences a loss of regularity (usually observed for the case of coefficients bounded in $x$) but also a decay in relation to the initial datum defined in a Sobolev space tailored to the generalized symbol class. Further, we observe that a precise behavior of the solution could be obtained by making an optimal choice of the metric in relation to the coefficients of the given equation. We also derive the cone conditions in the global setting.
