Asymptotic behaviour of the finite blow-up points solutions of the fast diffusion equation

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in {\mathbb{Z}}^{+}$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a smooth bounded domain, $a_1,a_2,\dots ,a_{i_0}\in \mathrm{\Omega }$, $\widehat{\mathrm{\Omega }}=\mathrm{\Omega }\setminus \{a_1,a_2,\dots ,a_{i_0}\}$, $0\le f\in L^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })$ and $0\le u_0\in L_{loc}^p(\widehat{\mathrm{\Omega }})$ for some constant $p>\frac{n(1-m)}{2}$ which satisfies ${\lambda }_i\mathrm{\mid }x-a_i{\mathrm{\mid }}^{-{\gamma }_i}\le u_0(x)\le {\lambda }_i^{\mathrm{\prime }}\mathrm{\mid }x-a_i{\mathrm{\mid }}^{-{\gamma }_i^{\mathrm{\prime }}}\mathrm{\forall }0<\mathrm{\mid }x-a_i\mathrm{\mid }<\delta$, $i=1,\dots ,i_0$ where $\delta >0$, ${\lambda }_i^{\mathrm{\prime }}\ge {\lambda }_i>0$ and $\frac{2}{1-m}<{\gamma }_i\le {\gamma }_i^{\mathrm{\prime }}<\frac{n-2}{m}$ $\mathrm{\forall }i=1,2,\dots ,i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=\mathrm{\Delta }{u}^m$ in $\widehat{\mathrm{\Omega }}\times (0,\mathrm{\infty })$, $u(a_i,t)=\mathrm{\infty }\mathrm{\forall }i=1,\dots ,i_0,t>0$, $u(x,0)=u_0(x)$ in $\widehat{\mathrm{\Omega }}$ and $u=f$ on $\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty })$, as $t\to \mathrm{\infty }$. We will construct finite blow-up points solution in bounded cylindrical domain with appropriate lateral boundary value such that the finite blow-up points solution oscillates between two given harmonic functions as $t\to \mathrm{\infty }$. We will also prove the existence of the minimal solution of $u_t=\mathrm{\Delta }{u}^m$ in $\widehat{\mathrm{\Omega }}\times (0,\mathrm{\infty })$, $u(x,0)=u_0(x)$ in $\widehat{\mathrm{\Omega }}$, $u(a_i,t)=\mathrm{\infty }\mathrm{\forall }t>0,i=1,2\dots ,i_0$ and $u=\mathrm{\infty }$ on $\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty })$.