Abstract:In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time $t=0$. Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtained the scattering result and its asymptotic order for the data in the energy space $H^1\times L^2$ when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on $r$ when the initial data is in $L^r$ ($1\leq r < 2$). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for $L^r$-data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

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中文翻译：

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用𝐿^{𝑟}-data 的尺度不变阻尼和质量的线性波动方程的渐近阶注

摘要：在本文中，我们考虑了具有尺度不变阻尼和质量的线性波动方程。众所周知，解的全局行为取决于初始时间 $t=0$ 时阻尼和质量前面的系数的大小。实际上，如果解很大，则解满足与相应热方程相似的衰减估计，如果小，则满足修正波动方程的类似衰减估计。在我们之前的论文中，我们得到了能量空间$H^1\times L^2$当系数处于波浪状态时数据的散射结果及其渐近阶次。事实上，系数的阈值依赖于初始数据的空间衰减。即，当初始数据在 $L^r$ ($1\leq r < 2$) 中时，它取决于 $r$。在本文中，我们将展示$L^r$-数据在波态中的散射结果和渐近阶数，它比能量空间中数据的波态更宽。此外，我们对之前论文中获得的能量空间数据的渐近阶进行了改进。

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