Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912527436218368 |
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| author | Iagar, Razvan Gabriel Laurençot, Philippe |
| author_facet | Iagar, Razvan Gabriel Laurençot, Philippe |
| contents | Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$ \partial\_tu=Δu^m+|x|^σu^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $σ>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^γu\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that non-existence of solution occurs for $$ γ<\frac{σ+2}{p-m} - \frac{2\max{\{p-p\_G,0\}}}{(p-1)(p-m)} $$ with $$ p\_G:=1+\frac{σ(1-m)}{2}. $$ The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of $γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06164 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion Iagar, Razvan Gabriel Laurençot, Philippe Analysis of PDEs Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$ \partial\_tu=Δu^m+|x|^σu^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $σ>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^γu\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that non-existence of solution occurs for $$ γ<\frac{σ+2}{p-m} - \frac{2\max{\{p-p\_G,0\}}}{(p-1)(p-m)} $$ with $$ p\_G:=1+\frac{σ(1-m)}{2}. $$ The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of $γ$. |
| title | Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.06164 |