Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910252351356928 |
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| author | Fang, Fei Tan, Zhong |
| author_facet | Fang, Fei Tan, Zhong |
| contents | In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*}
\begin{cases}u_t-Δu-μ\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-Δv-\frac{1}{2} y \cdot \nabla v=βv+\frac{μv}{|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 $N \geqslant 3$, $μ\in [0,(N-2)^2 /8]$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN Fang, Fei Tan, Zhong Analysis of PDEs In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*} \begin{cases}u_t-Δu-μ\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-Δv-\frac{1}{2} y \cdot \nabla v=βv+\frac{μv}{|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 $N \geqslant 3$, $μ\in [0,(N-2)^2 /8]$ 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. |
| title | Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.24802 |