Existence of new self-similar solutions of the fast diffusion equation

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\eta>0$, $\eta_0>0$, $\rho_1>0$, $-\frac{\rho_1}{2}<\beta<\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. We will prove the existence of radially symmetric solution of the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$, which satisfies $f(0)=\eta_0$, $f_r(0)=0$. When $\beta<\frac{m\rho_1}{n-2-nm}$ holds instead, we will also prove the existence of radially symmetric solution of the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, which satisfies $\lim_{x\to\infty}|x|^{\frac{n-2}{m}}f(x)=\eta$. As a consequence if $f_1$, $f_2$, are the solutions of the above two problems with $\rho_1=1$, then the function $V_i(x,t)=(T-t)^{\alpha}f_i(T-t)^{\beta} x)$, $i=1,2$, are backward similar solutions of the fast diffusion equation $u_t=\Delta (u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$ and $(\mathbb{R}^n\setminus\{0\})\times (-\infty,T)$ respectively.