# [Differential Equations II] Final Report

Edited by Leun Kim

In this report, we study the global existence of solutions to nonlinear dispersive wave equations \begin{eqnarray*}\partial_t^2 u + \frac{1}{\rho^2} |\partial_x|^{2\rho} u &=& \lambda |\partial_t u |^{p-1} \partial_t u\end{eqnarray*} in one space dimension where $0 < \rho \leqslant 2$, $\rho \neq 1$, $p >3$ and $\lambda \in \mathbb C$.

In the second part, we introduce some $L^1$-$L^\infty$ decay estimates for some partial differential equations as well as Schrödinger equations in $n$ space dimension. For the Schrödinger equations, we extended the $L^1$-$L^\infty$ estimate for the Schrödinger equations to $n$ space dimension by the similar method in Hayashi [1], [2] for one space dimension which uses the free evolution group. And for the wave equations and the Klein-Gordon equations, the proof is similar to Wahl [3] which is given by the classical methods.
In the third part, we will prove the global existence of solutions to the nonlinear dispersive wave equations above which is given by Hayashi [2] using some estimates for the free evolution group. We will give some more details of Hayashi [2] there.

Hayashi_Report_20130128

[1] N. Hayashi, P. I. Naumkin. Asymptotic Properties of Solutions to Dispersive Equation of Schrödinger Type. J. Math. Soc. Japan, Vol.60, No.3, pp.631-652, 2008.
[2] N. Hayashi, S. Kobayashi, P. I. Naumkin. Global Existence of Solutions to Nonlinear Dispersive Wave Equations. Differential and Integral Equations, Vol.25, No.7-8, pp.685-698, 2012.
[3] W. von Wahl. $L^p$-Decay Rates for Homogeneous Wave Equations. Math. Z, 120, pp.93-106, 1971.

#### Leun Kim

Ph.D Candidate at The University of Tokyo
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.