Existence of sign-changing bubble towers for the critical heat equation with drift term

This paper deals with the following Cauchy problem for critical heat equation with drift term $ \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u +\nabla \ln a(x)\cdot \nabla u +|u|^{\frac{4}{n-2}}u, & \mbox{for}\ \ &(x, t)\in \mathbb{R}^n\times (0, +\infty), \\ &u(\cdot, 0) = u_0, & \mbox{for}\ \ &x\in \mathbb{R}^n, \end{aligned} \right. \end{equation*} $ where $ a(x) $ is a positive smooth bounded function in $ \mathbb{R}^n, n\geq 6 $. Assume that the eigenvalues of matrix $ A(q) $, denoted as $ \sigma_i, \; i = 1, \cdots, n $, satisfy$ \begin{equation*} \sigma_i<(1+\sigma)\frac{3n}{n+2}\sum\limits_{i = 1}^{n}\frac{\sigma_i}{n}. \end{equation*} $We construct sign-changing bubble tower solutions which blow up at the critical point $ q $ of $ a(x) $ in the forward time infinity of the form$ u(x, t)\approx \sum\limits_{j = 1}^{2}(-1)^{j-1}\alpha_n \bigg(\frac{\mu_j}{\mu_j^2+|x-q|^2}\bigg)^{\frac{n-2}{2}} \ \ \text{as}\ \ t\rightarrow +\infty, $here $ 0<\mu_j(t)\rightarrow 0 $ exponentially as $ t\rightarrow +\infty. $