Properties of Pseudodifferential Operators and Their Applications to Hyperbolic Equations
Project/Area Number  01540151 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
解析学

Research Institution  University of Osaka Prefecture 
Principal Investigator 
TANIGUCHI Kazuo Instructor, 総合科学部, 講師 (80079037)

CoInvestigator(Kenkyūbuntansha) 
TOZAKI Yoshiharu Assistant, 総合科学部, 助手 (70079036)
SHINKAI Kenzo Professor, 総合科学部, 教授 (50079034)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1990 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1989 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  hyperbolic equation / ultra wave front set / Gevrey class / propagation of singularity / Cauchy problem / Fourier integral operator / complex phase function / pourus media equation / 双曲型方程式 / 超波面集合 / ジェヴレイ関数族 / 特異性の伝播 / 初期値問題 / フリェ積分作用素 / 複素相関数 / ポラスメディア方程式 / フリエ積分作用素 / ジェヴレイ関数 / 波面集合 / 擬微分作用素 
Research Abstract 
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^* wellposed, the symbol of the fundamental solution has exponential growth, that is, it is estimated not only from above but also from below by (1) <numerical formula> The constant kappa in (1) corresponds to the constant in the necessary and sufficient condition for the wellposedness in Gevrey classes. In order to study this phenomenon we define UWF^<(mu)>(u) (ultra wave front sets) for u that belongs to the space of ultradist ributions S{kappa}' by <numerical formula> where X * S{kappa}*C^*_ and xi belongs to a conic neighborhood of xi_<omicron>. Then by using UWF^<(mu)>(u) we can state the propagation of very high singularities for the solution of not C^* wellposed Cauchy problem. We also construct the ndamental solutions of the Cauchy problem for degenerate hyperbolic operators (2) <numerical formula> (3) <numerical formula> with an even integer j and we investigated other related topics.

Report
(4results)
Research Output
(13results)