L^1-L^∞ Decay Estimates for the Odd-Dimensional Wave Equations

Edited by Leun Kim

We will show the following estimates.

For $n = 1,3,\cdots$, let $\phi \in C_0^{n} (\Bbb R^n), \; \psi \in C_0^{n-1} (\Bbb R^n) $. And let $v$ solve
\begin{equation}\partial_t^2 v – \Delta v = 0 \;\text{in}\; [0,\infty) \times \Bbb R^n \;\; \text{ with } \;\; v(t=0) = \phi, \; \partial_t v (t=0) = \psi.\end{equation} Then there exists $c = c(n)>0$ such that for $t \geqslant 0$,
\begin{equation} \| v(t) \|_\infty \leqslant c(1+t)^{- \frac{n-1}{2}} ( \| \phi \|_{n,1} + \| \psi \|_{n-1,1} ).

In fact, for the even dimensions, the similar estimates hold. I will post this later with the case of non-vanishing mass for the Klein-Gordon equation.




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.