Elucidation of the geometric and analytic structure of Schroedinger equations on symmetric spaces and its applications

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400116
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: University of Tsukuba (2016-2017); Okayama University (2014-2015)
• Principal Investigator: KAKEHI Tomoyuki 筑波大学, 数理物質系, 教授 (70231248)
• Co-Investigator(Kenkyū-buntansha): 田村 英男 岡山大学, 自然科学研究科, 特命教授 (30022734)
• Co-Investigator(Renkei-kenkyūsha): KIYOHARA Kazuyoshi 岡山大学, 自然科学研究科, 教授 (80153245) ; YAMADA Hirofumi 熊本大学, 大学院先端科学研究部(理), 教授 (40192794)
• Research Collaborator: GONZALEZ Fulton タフツ大学, 数学教室, 教授
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 対称空間 / シュレディンガー方程式 / 基本解 / 幾何解析 / ガウス和

Outline of Final Research Achievements

In this research, we studied the following two subjects (A) and (B) which are closely related to geometric analysis of Schoedinger equations on symmetric spaces.(A) Mean value operators on symmetric spaces. (B) A certain reaction-diffusion system with the fractional Laplacian. Briefly our results are as follows.(A) We proved that under some conditions the mean value operator is surjective as an operator on the space of smooth functions on noncompact symmetric spaces. (B) We proved the existence of a global in time solution under some conditions on the nonlinear terms. We also determined the critical exponent for blowup of the solution. Moreover, we gave the optimal estimate for the lifespan of the blowup solution.

Free Research Field

微分方程式