Recently, by using the argument of Lei & Lin (2011) [11], Liu & Gao (2017) [13] establish the global well-posedness of mild solutions to the three-dimensional Boussinesq equations in the space chi(-1) defined by chi(-1) = {u is an element of D'(R-3) : integral(R3) vertical bar xi vertical bar(-1)vertical bar(xi) over cap (-1)vertical bar xi < infinity < col. However, it seems that their proof is incorrect, and has some obvious and essential mistakes. Compared with the Navier-Stokes equations, it is difficulty to obtain a global well-posedness of mild solutions to the Boussinesq system in the space chi(-1). In this paper, we will point out the mistakes of Liu Sz Gao. And, furthermore, in order to understand the difficulty of the Boussinesq system better, we study an illuminating system as follows: {partial derivative(t)u + (u . del)u - mu(1 + t)(alpha) del u + del p = theta e(3), in R-3 x (0, infinity), partial derivative(t)theta + (u . del)theta - k (1 + t)(alpha) Delta theta, in R-3 x (0, infinity), del . u = 0, in R-3 x (0, infinity), u(x, 0) = u(0), theta(x,0) = theta degrees, in R-3, where mu > 0, k > 0 and alpha > 1 are real constant parameters. By using the time-weighted estimate, we can show that the above system has a global mild solution. (C) 2019 Elsevier Inc. All rights reserved.