Long time behavior of solutions of nonlinear dispersive equations

我解决了几个问题。我证明了一些具有散焦(即排斥)非线性的勉强超临界薛定谔方程的整体非光滑解的存在性和散布：这导致了最近发表在国际数学研究通知上的一篇论文。我还证明了具有聚焦(即吸引)非线性的依赖于大小的勉强超临界薛定谔方程的整体光滑解和基态下非光滑解的散布：这导致了预印本(arxiv：1701.02689)。我还与久川幸太郎教授(名古屋大学和中央大学)一起研究了一个问题，该问题涉及构造完全非线性方程的局部光滑解，该方程满足弱意义下的色散性质。这一合作即将导致预印本的发布。

I worked on several problems. I proved global existence and scattering of nonsmooth solutions of some barely supercritical Schrodinger equations with a defocusing (i.e repulsive) nonlinearity: this lead to a paper recently published in International Mathematics Research Notices. I also proved global existence and scattering of smooth and nonsmooth solutions below ground state of some size-dependent barely supercritical Schrodinger equations with a focusing (i.e attractive) nonlinearity: this lead to a preprint (arXiv:1701.02689). I also worked on a problem with Prof Kotaro Tsugawa (Nagoya University and Chuo University) that involves the construction of local smooth solutions of fully nonlinear equations that satisfya dispersive property in a weak sense. This collaboration is about to lead to a preprint.