On the Hardy-Hénon heat equation with an inverse square potential
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| Format: | Preprint |
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2024
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| author | Bhimani, Divyang G. Haque, Saikatul Ikeda, Masahiro |
| author_facet | Bhimani, Divyang G. Haque, Saikatul Ikeda, Masahiro |
| contents | We study Cauchy problem for the Hardy-Hénon parabolic equation with an inverse square potential, namely, \[\partial_tu -Δu+a|x|^{-2} u= |x|^γ F_α(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $γ\in \mathbb R$, $α>1$ and $F_α(u)=μ|u|^{α-1}u, μ|u|^α$ or $μu^α$, $μ\in \{-1,0,1\}$. We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t (-Δ+ a|x|^{-2})}$ in weighted Lebesgue spaces. This may be of independent interest.
As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on $γ$ and $α,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for $α>1+\frac{2+γ}{d}$ the Fujita exponent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hardy-Hénon heat equation with an inverse square potential Bhimani, Divyang G. Haque, Saikatul Ikeda, Masahiro Analysis of PDEs 35K05, 35K67, 35B30, 35K57 We study Cauchy problem for the Hardy-Hénon parabolic equation with an inverse square potential, namely, \[\partial_tu -Δu+a|x|^{-2} u= |x|^γ F_α(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $γ\in \mathbb R$, $α>1$ and $F_α(u)=μ|u|^{α-1}u, μ|u|^α$ or $μu^α$, $μ\in \{-1,0,1\}$. We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t (-Δ+ a|x|^{-2})}$ in weighted Lebesgue spaces. This may be of independent interest. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on $γ$ and $α,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for $α>1+\frac{2+γ}{d}$ the Fujita exponent. |
| title | On the Hardy-Hénon heat equation with an inverse square potential |
| topic | Analysis of PDEs 35K05, 35K67, 35B30, 35K57 |
| url | https://arxiv.org/abs/2407.13085 |