ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS |
| |
Authors: | ZHOU Yi |
| |
Institution: | Institute of Mathematics, Fudan University, Shanghai 200433, China. |
| |
Abstract: | This paper considers the following
Cauchy problem for semilinear wave equations in $n$ space
dimensions
$$\align
\square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x),
\endalign$$
where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is
quadratic in $\partial\p$ with
$\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$.
The minimal value of $s$ is determined such that the above
Cauchy problem is locally well-posed in $H^s$. It turns out that
for the general equation $s$ must satisfy
$$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$
This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge
5$). The purpose of this paper is to supplement with a proof in the
case $n=2,4$. |
| |
Keywords: | Semilinear wave equation Cauchy problem Low regularity solution |
|
| 点击此处可从《数学年刊A辑(中文版)》浏览原始摘要信息 |
| 点击此处可从《数学年刊A辑(中文版)》下载免费的PDF全文 |
|