Asymptotic stability of homogeneous solutions to Navier-Stokes equations under $L^p$-perturbations

It is known that there has been classified for all $(-1)$-homogeneous axisymmetric no-swirl solutions of the three-dimensional Navier-Stokes equations with a possible singular ray. The main purpose of this paper is to show that the least singular solutions among such solutions other than Landau solutions to the Navier-Stokes equations are asymptotically stable under $L^3$-perturbations. Moreover, we establish the $L^q$ decay estimate with an explicit decay rate and a sharp constant for any $q>3$. For that purpose, we first study the global well-posedness of solutions to the perturbed equations under small initial data in $L_{\sigma }^3$ space and the local well-posedness with any initial data in $L_{\sigma }^p$ spaces for $p\ge 3$.