On solutions of critical nonlinear dispersive and dissipative equations

2019 Fiscal Year Final Research Report

• Project/Area Number: 15H03630
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Osaka University
• Principal Investigator: Hayashi Nakao 大阪大学, 理学研究科, 教授 (30173016)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: Nonlinear Schredinger / Boundary value problem / Scattering problem / Asymtotic behavior / Critical nonlinearity / Invariant space / Nonlinear dispersive / Nonlinear dissipative

Outline of Final Research Achievements

We studied the initial value problem for nonlinear dispersive equations including Critical nonlinear Schredinger equations in higher space dimensions, Fractional order nonlinear Schredinger equations with a critical nonlinearity, Nonlinear disipative wave equations, Higher order nonlinear Schredinger equations. We also studied inhomogeneous initial baoundary value problem for nonlinear Schredinger equations. For these five years, we showed asymptotic behavior of solutions, scattering problem of solutions, global existence in time of solutions in a scale invariant space for these nonlinear problems. We published 25 papers including our results in international journals. We also presented our results in mathematical meetings held in Japan and other countries as possible as we can.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

高階非線形Schredinger方程式, 分数冪非線形Schredinger方程式の研究において,因数分解公式の方法が有効であることを示すことができたこと,高次元臨界冪非線形Schredinger方程式, 高次元非線形消散型波動の時間大域解の存在, 解の漸近的振る舞いに関する成果は当該研究分野の発展に寄与した. 特にSchredinger方程式の研究において従来の関数空間とは異なる空間を利用したことは学術的に意義があると考える. また非線形Schredinger方程式の非斉次境界値問題に関する研究は従来活発に行われていなかった研究で, 得られた成果によりさらなる活性化が期待できる.