Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions
Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions
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DOI：
10.1016/j.jde.2013.04.003
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发表时间：
2012-05
影响因子：
2.4
通讯作者：
S. Katayama
中科院分区：
文献类型：
--
作者：
S. Katayama
本文研究三维空间中具有多重传播速度的非线性波动方程组的柯西问题。在零条件下，我们得到了小振幅解的整体存在性。在本文中，我们将通过获得解的导数的渐近逐点行为，证明整体解在能量意义下是渐近自由的。尽管如此，我们也可以证明解本身的逐点行为可能与自由解的逐点行为完全不同。结合上述结果，我们还得到了一个定理，用以刻画任意空间维中波动方程的渐近自由解。
We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In this paper, we will show that the global solution is asymptotically free in the energy sense, by obtaining the asymptotic pointwise behavior of the derivatives of the solution. Nonetheless we can also show that the pointwise behavior of the solution itself may be quite different from that of the free solution. In connection with the above results, a theorem is also developed to characterize asymptotically free solutions for wave equations in arbitrary space dimensions.