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| ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS |
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Citation： |
ZHOU Yi.ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS[J].Chinese Annals of Mathematics B,2003,24(3):293~302 |
| Page view： 1200
Net amount： 1275 |
Authors： |
ZHOU Yi; |
Foundation： |
Project supported by the 973 Project of the National Natural Science Foundation of China, the Key
Teachers Program and the Doctoral Program Foundation of the Ministry of Education of China. |
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| 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 |
Classification： |
35L |
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