Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2021-05-17 , DOI: 10.1007/s00028-021-00701-6 María E. Martínez
We are interested in the long time asymptotic behaviour of solutions to the scalar Zakharov system
$$\begin{aligned} \begin{array}{ll} i u_{t} + \Delta u = nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$and the Klein–Gordon–Zakharov system
$$\begin{aligned} \begin{array}{ll} u_{tt} - \Delta u + u = - nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals assuming smallness and parity conditions (u odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an increasing \(C^1\) function. No smallness condition is needed to prove this last result for the Zakharov system. We argue relying on the use of suitable virial identities appropriate for the equations and follow the technics of [22, 24] and [33].
中文翻译：
一维Zakharov和Klein-Gordon-Zakharov系统的衰减问题
我们对标量Zakharov系统解的长时间渐近行为感兴趣
$$ \ begin {aligned} \ begin {array} {ll} i u_ {t} + \ Delta u = nu，\\ n_ {tt}-\ Delta n = \ Delta | u | ^ 2 \ end {array} \ end {aligned} $$以及Klein-Gordon-Zakharov系统
$$ \ begin {aligned} \ begin {array} {ll} u_ {tt}-\ Delta u + u =-nu，\\ n_ {tt}-\ Delta n = \ Delta | u | ^ 2 \ end {数组} \ end {aligned} $$在空间的一维。对于这两个系统，我们给出两个结果证明能量空间中初始数据的解衰减。第一个结果假定小和奇偶条件（u奇数），处理紧致间隔上的衰减。第二个结果证明了对于远场区域，沿着曲线的衰减对于其增长可以由递增的（C ^ 1 \）函数决定的解。对于Zakharov系统，不需要任何小巧的条件即可证明最后的结果。我们主张依靠适合方程的适当病毒身份，并遵循[22，24]和[33]的技术。
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