Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909568003473408 |
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| author | Hsu, Shu-Yu |
| author_facet | Hsu, Shu-Yu |
| contents | Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $γ>0$ and $η>0$. Suppose either (i) $α\ne 0$ and $β=0$ or (ii) $α\in\mathbb{R}$ and $β\ne 0$ holds. We will study the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$ with $\underset{\substack{r\to 0}}{\lim}\,r^γf(r)=η$. This equation arises from the study of the singular self-similar solutions of the fast diffusion equation which blow up at the origin. We will prove that if there exists a radially symmetric singular solution of the above elliptic equation, then either $γ=\frac{2}{1-m}$ and $α>\frac{2β}{1-m}$ or $γ>\frac{2}{1-m}$, $β\ne 0$ and $γ=α/β$. As a consequence we obtain the non-existence of radially symmetric self-similar solution of the fast diffusion equation $u_t=Δ(u^m/m)$, $u>0$, which blows up at the origin with rate $|x|^{-γ}$ when either $0<γ\ne\frac{2}{1-m}$ and $γ\neα/β$, $α\in\mathbb{R}$ and $β\ne 0$ or $γ=\frac{2}{1-m}$ and $\left(α-\frac{2β}{1-m}\right)η\ne\frac{2(n-2-nm)}{(1-m)^2}$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation Hsu, Shu-Yu Analysis of PDEs AMS 2020 Mathematics Subject Classification: Primary 35K65, 35J70 Secondary 35B09, 35B44 Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $γ>0$ and $η>0$. Suppose either (i) $α\ne 0$ and $β=0$ or (ii) $α\in\mathbb{R}$ and $β\ne 0$ holds. We will study the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$ with $\underset{\substack{r\to 0}}{\lim}\,r^γf(r)=η$. This equation arises from the study of the singular self-similar solutions of the fast diffusion equation which blow up at the origin. We will prove that if there exists a radially symmetric singular solution of the above elliptic equation, then either $γ=\frac{2}{1-m}$ and $α>\frac{2β}{1-m}$ or $γ>\frac{2}{1-m}$, $β\ne 0$ and $γ=α/β$. As a consequence we obtain the non-existence of radially symmetric self-similar solution of the fast diffusion equation $u_t=Δ(u^m/m)$, $u>0$, which blows up at the origin with rate $|x|^{-γ}$ when either $0<γ\ne\frac{2}{1-m}$ and $γ\neα/β$, $α\in\mathbb{R}$ and $β\ne 0$ or $γ=\frac{2}{1-m}$ and $\left(α-\frac{2β}{1-m}\right)η\ne\frac{2(n-2-nm)}{(1-m)^2}$ holds. |
| title | Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation |
| topic | Analysis of PDEs AMS 2020 Mathematics Subject Classification: Primary 35K65, 35J70 Secondary 35B09, 35B44 |
| url | https://arxiv.org/abs/2503.08448 |