Fractional heat equation involving Hardy-Leray Potential
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929244339175424 |
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| author | Abdellaoui, Boumediene Siclari, Giovanni Primo, Ana |
| author_facet | Abdellaoui, Boumediene Siclari, Giovanni Primo, Ana |
| contents | In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-Δw)^s=\fracλ{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases} $$ where $N> 2s$, $0<s<1$ and $0<λ<Λ_{N,s}$, the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent $p_{+}(λ, s)$ and of a Fujita-type exponent $F(λ,s)$ such that the following holds:
- Let $p>p_+(λ,s)$. Then there are not any non-negative supersolutions.
- Let $p<p_+(λ,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases:
- Let $ 1< p\le F(λ,s)$. Here we show that a weighted norm of any positive solution blows up in finite time.
- Let $F(λ,s)<p<p_+(λ,s)$. Here we prove the existence of global solutions under suitable hypotheses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional heat equation involving Hardy-Leray Potential Abdellaoui, Boumediene Siclari, Giovanni Primo, Ana Analysis of PDEs 35B25, 35B44, 35K58, 35B33, 47G20 In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-Δw)^s=\fracλ{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases} $$ where $N> 2s$, $0<s<1$ and $0<λ<Λ_{N,s}$, the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent $p_{+}(λ, s)$ and of a Fujita-type exponent $F(λ,s)$ such that the following holds: - Let $p>p_+(λ,s)$. Then there are not any non-negative supersolutions. - Let $p<p_+(λ,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let $ 1< p\le F(λ,s)$. Here we show that a weighted norm of any positive solution blows up in finite time. - Let $F(λ,s)<p<p_+(λ,s)$. Here we prove the existence of global solutions under suitable hypotheses. |
| title | Fractional heat equation involving Hardy-Leray Potential |
| topic | Analysis of PDEs 35B25, 35B44, 35K58, 35B33, 47G20 |
| url | https://arxiv.org/abs/2402.09862 |