Asymptotic large time behavior of singular solutions of the fast diffusion equation
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2025
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| author | Hui, Kin Ming Kim, Jongmyeong |
| author_facet | Hui, Kin Ming Kim, Jongmyeong |
| contents | Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $α=\frac{2β-1}{1-m}$ and $\frac{2}{1-m}<\fracαβ<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward self-similar solution of the form $V(x,t)=t^{-α} f(t^{-β}x)$ $\forall x\in\mathbb{R}^n\setminus\{0\}$, $t>0$, for the fast diffusion equation $u_t=Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} Δ(f^m/m) + αf + βx \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \fracαβ}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-γ}\leq u_0(x)\leq A_2|x|^{-γ}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\leγ<\frac{n-2}{m}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic large time behavior of singular solutions of the fast diffusion equation Hui, Kin Ming Kim, Jongmyeong Analysis of PDEs 35B35, 35B44, 35J70, 35K55, 35K65 Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $α=\frac{2β-1}{1-m}$ and $\frac{2}{1-m}<\fracαβ<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward self-similar solution of the form $V(x,t)=t^{-α} f(t^{-β}x)$ $\forall x\in\mathbb{R}^n\setminus\{0\}$, $t>0$, for the fast diffusion equation $u_t=Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} Δ(f^m/m) + αf + βx \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \fracαβ}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-γ}\leq u_0(x)\leq A_2|x|^{-γ}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\leγ<\frac{n-2}{m}$. |
| title | Asymptotic large time behavior of singular solutions of the fast diffusion equation |
| topic | Analysis of PDEs 35B35, 35B44, 35J70, 35K55, 35K65 |
| url | https://arxiv.org/abs/2506.11692 |