Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913507899867136 |
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| author | Fino, Ahmad Z. Kirane, Mokhtar Barakeh, Bilal Kerbal, Sebti |
| author_facet | Fino, Ahmad Z. Kirane, Mokhtar Barakeh, Bilal Kerbal, Sebti |
| contents | The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group $$ \begin{cases} u_{t}-Δ_{\mathbb{H}}u\geq (\mathcal{K}\ast_{_{\mathbb{H}}}|u|^p)|u|^q ,\qquad {η\in \mathbb{H}^n,\,\,\,t>0,}
\\{}\\ u(η,0)=u_{0}(η), \qquad\qquad\qquad\quad η\in \mathbb{H}^n,
\end{cases} $$ where $n\geq1$, $p,q>0$, $u_0\in L^1_{loc}(\mathbb{H}^n)$, $Δ_{\mathbb{H}}$ is the Heisenberg Laplacian, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function satisfying $\mathcal{K}(|\cdotp|_{_{\mathbb{H}}})\in L^1_{loc}(\mathbb{H}^n)$ which decreases in a vicinity of infinity. In addition, $\ast_{_{\mathbb{H}}}$ denotes the convolution operation in $\mathbb{H}^n$. Our approach is based on the non-linear capacity method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group Fino, Ahmad Z. Kirane, Mokhtar Barakeh, Bilal Kerbal, Sebti Analysis of PDEs 35A01, 35R03, 35B53, 35B33, 35B45 The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group $$ \begin{cases} u_{t}-Δ_{\mathbb{H}}u\geq (\mathcal{K}\ast_{_{\mathbb{H}}}|u|^p)|u|^q ,\qquad {η\in \mathbb{H}^n,\,\,\,t>0,} \\{}\\ u(η,0)=u_{0}(η), \qquad\qquad\qquad\quad η\in \mathbb{H}^n, \end{cases} $$ where $n\geq1$, $p,q>0$, $u_0\in L^1_{loc}(\mathbb{H}^n)$, $Δ_{\mathbb{H}}$ is the Heisenberg Laplacian, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function satisfying $\mathcal{K}(|\cdotp|_{_{\mathbb{H}}})\in L^1_{loc}(\mathbb{H}^n)$ which decreases in a vicinity of infinity. In addition, $\ast_{_{\mathbb{H}}}$ denotes the convolution operation in $\mathbb{H}^n$. Our approach is based on the non-linear capacity method. |
| title | Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group |
| topic | Analysis of PDEs 35A01, 35R03, 35B53, 35B33, 35B45 |
| url | https://arxiv.org/abs/2409.12508 |