Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*} \begin{cases}u_t-\Delta u-\mu \frac{u}{|x|^2}=|u|^{2^*-2} u & \text { in } \mathbb{R}^N \times(0, T), u(x, 0)=u_0(x) & \text { in } \mathbb{R}^N , \end{cases} \end{equation*} into the following parabolic equation \begin{equation*} \begin{cases} v_s-\Delta v-\frac{1}{2} y \cdot \nabla v=\beta v+\frac{\mu v}{|y|^2}+|v|^{2^*-2} v &\text { in } \mathbb{R}^N \times(0, S), \left.v\right|_{s=0}=v_0 & \text { in } \mathbb{R}^N, \end{cases} \end{equation*} where $N \geqslant 3$, $\mu\in [0,(N-2)^2 /8]$ and $2^{\ast}=2N /(N-2)$. For this equation, we establish a weighted Hardy inequality. Furthermore, by virtue of the modified potential well method and Palais-Smale sequence analysis, we investigate the long-time behavior and finite-time blow-up properties of solutions to the parabolic equation.