Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation

Abstract

Let $n\ge 3$, $0 0$, $η>0$, $β>\frac{mρ_1}{n-2-nm}$, $α=α_m=\frac{2β+ρ_1}{1-m}$, $β_0>0$ and $α_0=2β_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}f(x)=η$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $Δ\log g+α_0 g+β_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α_0/β_0}g(x)=η$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=Δu^m$ and the logarithmic diffusion equation $u_t=Δ\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.