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

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, ${\rho }_1>0$, $\eta >0$, $\beta >\frac{m{\rho }_1}{n-2-nm}$, $\alpha ={\alpha }_m=\frac{2\beta +{\rho }_1}{1-m}$, ${\beta }_0>0$ and ${\alpha }_0=2{\beta }_0+1$. 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 $\Delta (f^m/m)+\alpha f+\beta x\cdot \nabla f=0$, $f>0$, in ${\mathbb{R}}^n\setminus \{0\}$, satisfying $\underset{|x|\to 0}{\lim }|x{|}^{\alpha /\beta }f(x)=\eta$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $\Delta \log g+{\alpha }_{0}g+{\beta }_{0}x\cdot \nabla g=0$, $g>0$, in ${\mathbb{R}}^n\setminus \{0\}$, satisfying $\underset{|x|\to 0}{\lim }|x{|}^{{\alpha }_0/{\beta }_0}g(x)=\eta$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=\Delta u^m$ and the logarithmic diffusion equation $u_t=\Delta \log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to \infty$.

35 pages, introduction rewritten and Theorem 1.1 and Theorem 1.2 combined into Theorem 1.1, Theorem 1.3 and Theorem 1.4 combined into Theorem 1.2