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

Kin Ming Hui

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 fo…