[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.

 

 
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.



2 comments

  • ㅎㅎㅎ 엄청 빡쎈 숙제네요.. 역시 Hayashi 센센이신가..;; 전에 Ponce교수 생일기념 학회에서 얘기나눴을때도 엄청 깐깐해 보이시던데..ㅋㅋㅋ 아 아직 먼얘기지만 3월에 오사카에서 열리는 Suzuki 센세 생일 기념 학회에 특별한일이 없으면 가게 될거 같은데 혹시 참석하나요?^^ 한국에서 채동호 교수님이랑 최희준 교수님도 오신다는거 같던데요..ㅎㅎㅎ

    • 매주 자신의 최신 논문들을 소개하는 식으로 수업을 꾸려나가시면서 학생들을 멘붕시켰었죠 ㅎㅎ 그 세미나에 참석하시는군요. 예전에 포스터를 통해 알고는 있었습니다만 참석할 지는 잘 모르겠군요. 일단 공식 일정이 끝나는대로 잠시 한국으로 귀국해서 부서진 앞니를 부활시킬 예정이라서요 ㅎㅎ