Blow-up and global mild solutions for a Hardy-Hénon parabolic equation on the Heisenberg group
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| Format: | Preprint |
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2025
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| author | Castillo, Ricardo Freire, Ricardo Loayza, Miguel |
| author_facet | Castillo, Ricardo Freire, Ricardo Loayza, Miguel |
| contents | We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-Hénon equation $u_t - Δ_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^γ u^p \mbox{ in } \mathbb{H}^N \times (0,T),$ where $\mathbb{H}^N$ is the $N$-dimensional Heisenberg group, and the singular term $|\cdot|_{\mathbb{H}}^γ$ is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for $γ\geq 0$, the Fujita critical exponent is $p_c = 1+ (2+γ)/Q$, where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$. For $γ<0$, the solutions blow up for $1<p<1+ (2+γ)/Q$, while global solutions exist for $p>1+ (2+γ)/(Q + γ)$. In particular, our results coincide with the results found by Georgiev and Palmieri in \cite{PALMIERI} for $γ=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up and global mild solutions for a Hardy-Hénon parabolic equation on the Heisenberg group Castillo, Ricardo Freire, Ricardo Loayza, Miguel Analysis of PDEs 35A01, 35B44, 35K55, 35K08, 35R03 We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-Hénon equation $u_t - Δ_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^γ u^p \mbox{ in } \mathbb{H}^N \times (0,T),$ where $\mathbb{H}^N$ is the $N$-dimensional Heisenberg group, and the singular term $|\cdot|_{\mathbb{H}}^γ$ is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for $γ\geq 0$, the Fujita critical exponent is $p_c = 1+ (2+γ)/Q$, where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$. For $γ<0$, the solutions blow up for $1<p<1+ (2+γ)/Q$, while global solutions exist for $p>1+ (2+γ)/(Q + γ)$. In particular, our results coincide with the results found by Georgiev and Palmieri in \cite{PALMIERI} for $γ=0$. |
| title | Blow-up and global mild solutions for a Hardy-Hénon parabolic equation on the Heisenberg group |
| topic | Analysis of PDEs 35A01, 35B44, 35K55, 35K08, 35R03 |
| url | https://arxiv.org/abs/2503.23208 |