Some remarks on large-time behaviors for the linearized compressible Navier-Stokes equations

In this paper, we consider the linearized compressible Navier-Stokes equations in the whole space ${\mathbb{R}}^n$. Concerning initial datum with suitable regularities, we introduce a new threshold $\mathrm{\mid }{\mathbb{B}}_0\mathrm{\mid }=0$ to distinguish different large-time behaviors. Particularly in the lower-dimensions, optimal growth estimates ($n=1$ polynomial growth, $n=2$ logarithmic growth) hold when $\mathrm{\mid }{\mathbb{B}}_0\mathrm{\mid }>0$, whereas optimal decay estimates hold when $\mathrm{\mid }{\mathbb{B}}_0\mathrm{\mid }=0$. Furthermore, we derive asymptotic profiles of solutions with weighted $L^1$ datum as large-time.