|
|
| |
| GLOBAL EXISTENCE FOR A CLASS OF SYSTEMS OF NONLINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS |
| |
Citation： |
S. KATAYAMA.GLOBAL EXISTENCE FOR A CLASS OF SYSTEMS OF NONLINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS[J].Chinese Annals of Mathematics B,2004,25(4):463~482 |
| Page view： 1080
Net amount： 890 |
Authors： |
S. KATAYAMA; |
|
|
| Abstract： |
Consider a system of nonlinear wave equations
$$
(\pa_t^2-c_i^2 \Delta_x)u_i=F_i(u,\pa u,\pa_x\pa u)
\qquad \text{in\ \ $(0, \infty)\times \R^3$}
$$
for $ i=1, \cdots, m $,
where $ F_i $ ($ i=1, \cdots, m$) are smooth functions of degree $
2 $ near the origin of their arguments, and $ u=(u_1, \cdots, u_m)
$, while $ \pa u $ and $ \pa_x \pa u $ represent the first and
second derivatives of $ u $, respectively. In this paper, the
author presents a new class of nonlinearity for which the global
existence of small solutions is ensured.
For example, global existence of small solutions for
$$
\begin{cases}
(\pa_t^2 -c_1^2 \Delta_x)u_1 = u_2(\pa_t u_2)
{}+\text{arbitrary cubic terms}, \\[1mm]
(\pa_t^2 -c_2^2 \Delta_x)u_2 = u_1(\pa_t u_2)+(\pa_t u_1)u_2%
{}+\text{arbitrary cubic terms}
\end{cases}
$$
will be established, provided that $ c_1^2 \ne c_2^2 $. |
Keywords： |
Wave equations, Multiple speeds, Global existence |
Classification： |
35L70 |
|
|
Download PDF Full-Text
|
|
|
|
