Abstract: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.
From: Fang Fei [view email]
[v1]
Sun, 24 May 2026 01:23:39 UTC (20 KB)
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