Decay of mass for a semilinear heat equation on Heisenberg group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915251971162112 |
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| author | Fino, Ahmad Z. |
| author_facet | Fino, Ahmad Z. |
| contents | In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_{t}-Δ_{\mathbb{H}}u=- k(t)u^p$ posed on $\mathbb{H}^n$, driven by the Heisenberg Laplacian and supplemented with a nonnegative integrable initial data, where $p>1$, $n\geq 1$, and $k:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+2/Q,$ while the classical/anomalous diffusion effects win if $p>1+{2}/{Q}$, where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14998 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of mass for a semilinear heat equation on Heisenberg group Fino, Ahmad Z. Analysis of PDEs 35K57, 35B40, 35R03, 35A01, 35B33 In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_{t}-Δ_{\mathbb{H}}u=- k(t)u^p$ posed on $\mathbb{H}^n$, driven by the Heisenberg Laplacian and supplemented with a nonnegative integrable initial data, where $p>1$, $n\geq 1$, and $k:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+2/Q,$ while the classical/anomalous diffusion effects win if $p>1+{2}/{Q}$, where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. |
| title | Decay of mass for a semilinear heat equation on Heisenberg group |
| topic | Analysis of PDEs 35K57, 35B40, 35R03, 35A01, 35B33 |
| url | https://arxiv.org/abs/2504.14998 |