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Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data

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.