Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908352040140800 |
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| author | Oza, Priyank Suragan, Durvudkhan |
| author_facet | Oza, Priyank Suragan, Durvudkhan |
| contents | In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-Δ_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita exponent, a critical threshold that delimits different dynamical regimes of this equation, is $$p_F\coloneqq\frac{Q}{Q-2s},$$ where $Q\coloneqq 2N+2$ is the homogeneous dimension of $\mathbbm{H}^N$. We prove the existence of global-in-time solutions for the supercritical case $(p>p_F),$ and the non-existence of global-in-time solutions for the subcritical case $(1<p<p_F).$ For the critical case $p=p_F,$ we provide a class of functions for which the solution blows up in finite time. These results extend the classical Fujita phenomenon to a sub-Riemannian setting with the nonlocal effects of the fractional sub-Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group Oza, Priyank Suragan, Durvudkhan Analysis of PDEs 35A01, 35R03, 47G20, 35B33, 35K58, 35B40 In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-Δ_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita exponent, a critical threshold that delimits different dynamical regimes of this equation, is $$p_F\coloneqq\frac{Q}{Q-2s},$$ where $Q\coloneqq 2N+2$ is the homogeneous dimension of $\mathbbm{H}^N$. We prove the existence of global-in-time solutions for the supercritical case $(p>p_F),$ and the non-existence of global-in-time solutions for the subcritical case $(1<p<p_F).$ For the critical case $p=p_F,$ we provide a class of functions for which the solution blows up in finite time. These results extend the classical Fujita phenomenon to a sub-Riemannian setting with the nonlocal effects of the fractional sub-Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group. |
| title | Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group |
| topic | Analysis of PDEs 35A01, 35R03, 47G20, 35B33, 35K58, 35B40 |
| url | https://arxiv.org/abs/2505.03619 |