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| Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations |
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Citation： |
Yi ZHOU,Wei HAN.Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations[J].Chinese Annals of Mathematics B,2011,32(4):521~526 |
| Page view： 2122
Net amount： 1911 |
Authors： |
Yi ZHOU; Wei HAN; |
Foundation： |
the National Natural Science Foundation of China (No. 10728101), the Basic
Research Program of China (No. 2007CB814800), the Doctoral Program Foundation of the Ministry of
Education of China, the “111” Project (No. B08018) and SGST (No. 09DZ2272900). |
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| Abstract： |
This paper is devoted to proving the sharpness on the lower bound of
the lifespan of classical solutions to general nonlinear wave
equations with small initial data in the case $n=2$ and cubic
nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992).
For this purpose, the authors consider the following Cauchy problem:
$$
\left\{\!\!\!
\begin{array}{l}
\Box u=(u_{t})^{3},\quad n=2, \t=0: \ u=0,\quad u_{t}=\varepsilon g(x),\quad x\in \mathbb{R}^{2},
\end{array}
\right.
$$
where
$\Box=\partial_{t}^{2}-\sum\limits_{i=1}^{n}\partial_{x_{i}}^{2}$ is
the wave operator, $g(x)\not\equiv 0$ is a smooth non-negative
function on $\mathbb{R}^{2}$ with compact support, and
$\varepsilon>0$ is a small parameter. It is shown that the solution
blows up in a finite time, and the lifespan $T(\varepsilon)$ of
solutions has an upper bound $T(\varepsilon) \leq \exp(A
\varepsilon^{-2})$ with a positive constant $A$ independent of
$\varepsilon$, which belongs to the same kind of the lower bound of
the lifespan. |
Keywords： |
Nonlinear wave equation, Cauchy problem, Lifespan |
Classification： |
35L45, 35L60 |
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