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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.24131 |
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Table of Contents:
- Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $η>0$, $η_0>0$, $ρ_1>0$, $-\frac{ρ_1}{2}<β<\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. We will prove the existence of radially symmetric solution of the equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$, which satisfies $f(0)=η_0$, $f_r(0)=0$. When $β<\frac{mρ_1}{n-2-nm}$ holds instead, we will also prove the existence of radially symmetric solution of the equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, which satisfies $\lim_{x\to\infty}|x|^{\frac{n-2}{m}}f(x)=η$. As a consequence if $f_1$, $f_2$, are the solutions of the above two problems with $ρ_1=1$, then the function $V_i(x,t)=(T-t)^αf_i(T-t)^β x)$, $i=1,2$, are backward similar solutions of the fast diffusion equation $u_t=Δ(u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$ and $(\mathbb{R}^n\setminus\{0\})\times (-\infty,T)$ respectively.