论文标题
关于线性波方程的渐近顺序的备注，并带有$ l^r $ -DATA的尺度不变性和质量
Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data
论文作者
论文摘要
在本文中，我们考虑了线性波方程，并具有尺度不变的阻尼和质量。众所周知，解决方案的全局行为取决于阻尼前面的系数的大小和初始时间$ t = 0 $的质量。实际上，如果溶液较大，则可以满足与相应热方程的相似衰减估计值，如果它很小，则可以满足改性波方程的相似衰减估计。在上一篇论文中，当系数处于波浪状态时，我们获得了能量空间中数据的散射结果及其渐近顺序。实际上，系数的阈值取决于初始数据的空间衰减。也就是说，当初始数据以$ l^r $（$ 1 \ leq r <2 $）为单位时，它取决于$ r $。在本文中，我们将以$ l^r $ -DATA的散射结果和波动状态中的渐近顺序，这比能量空间中数据的波状态宽。此外，我们改善了我们先前论文中有关能量空间中数据的渐近顺序。
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 obtain 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.
