Properties of Solutions of Partial Differential Equations and Their Applications

偏微分方程解的性质及其应用

基本信息

批准号：
63540134
负责人：
OKANO Hatsuo
金额：
$ 0.7万
依托单位：
University of Osaka Prefecture
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1988
资助国家：
日本
起止时间：
1988 至 1989
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-63540134/
关键词：
hyperbolic equation Fourier integral operator propagation of singularity Cauchy problem cohomology unitary representation class field lattice path combinatorics 波面集合コホモロジーユニタリー表現保型形式ブール関数 E.R.積分発散級数

项目摘要

The fundamental solution of the Cauchy problem for a hyperbolic operator is given in the form of Fourier integral operator. As shown below, when the problem is not C" well-posed, the symbol of the fundamental solution has exponential growth, that is, it is estimated not only from above but also from below by (1) C exp[cxi^<1/k>], c > 0, The constant kappa in (1) corresponds to the constant in the necessary and sufficient condition for the well-posedness in Gevrey classes. In order to study this phenomena we define UWF^<{mu}>(u)(ultra wave front sets) for u that belongs to the space of ultradistributions S{k}' by (chi_0,xi_0) <not a member of> UWF^<{mu}>(u) <tautomer> *_<epsilon> > O*C ; |X^u(xi)| <less than or equal> exp[epsilon < xi >^<1/mu>], where X * S{k}*C^*_ and xi belongs to a conic neighborhood of xi_0. Then by using UWP^<{mu}>(u) we can state the propagation of very high singularities for the solution of not C^* well-posed Cauchy problem. We also construct the fundamental solutions of the Cauchy problem for degenerate hyperbolic operators (2) L_1 = THETA^2_ - t^2_ - at^kTHETA^x with 0 < k < j - 1 and (3) L_2 = THETA^2_ - x^<2j>THETA^2_ - aTHETA^x with an even integer j and we investigated other related topics.

双曲线操作员的Cauchy问题的基本解决方案以傅立叶积分运算符的形式给出。 As shown below, when the problem is not C" well-posed, the symbol of the fundamental solution has exponential growth, that is, it is estimated not only from above but also from below by (1) C exp[cxi^<1/k>], c > 0, The constant kappa in (1) corresponds to the constant in the necessary and sufficient condition for the well-posedness in Gevrey classes. In order to study this phenomena we define uwf^<{mu}>（u）（utra Wave Front集）属于超级分布的空间S {k}'by（chi_0，xi_0） <xi>^<1/mu>]，其中x* s {k}* c^* _和xi属于XI_0的圆锥域，然后使用UWP^<{Mu}>（u），我们可以说出非常高的c^* cauchy问题的较高的单身人士的传播。 l_1 = theta^2_ -t^2_ -at^ktheta^x，带0 <k <j -1和（3）l_2 = theta^2_ -x^<2j> theta^2_ -atheta^2_ -atheta^x with Integer j with Integer j，我们研究了其他相关主题。