Asymptotic behavior for the fast diffusion equation with absorption and singularity

This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_t=\triangle u^m -u^p$. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0<m<1$ and the absorption index is $p>1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n-1}{n}<m<1$ and $p>m+\frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csisz$\mathrm{\acute{a}}$r-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.