Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a smooth bounded domain and let $a_1,a_2,\dots ,a_{i_0}\in \mathrm{\Omega }$, $\widehat{\mathrm{\Omega }}=\mathrm{\Omega }\setminus \{a_1,a_2,\dots ,a_{i_0}\}$ and $\widehat{R^n}={\mathbb{R}}^n\setminus \{a_1,a_2,\dots ,a_{i_0}\}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=\mathrm{\Delta }{u}^m$, $u>0$, in $\widehat{\mathrm{\Omega }}\times (0,\mathrm{\infty })$ ($\widehat{R^n}\times (0,\mathrm{\infty })$ respectively) which satisfies $u(x,t)\to \mathrm{\infty }$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots ,i_0$, when $0<m<\frac{n-2}{n}$, $n\ge 3$, and the initial value satisfies $0\le u_0\in L_{loc}^p(\text{2}\mathrm{\Omega }\setminus \{a_1,\cdots ,a_{i_0}\})$ ($u_0\in L_{loc}^p(\widehat{R^n})$ respectively) for some constant $p>\frac{n(1-m)}{2}$ and $u_0(x)\ge {\lambda }_i\mathrm{\mid }x-a_i{\mathrm{\mid }}^{-{\gamma }_i}$ for $x\approx a_i$andsomeconstants$\gamma_i>\frac{2}{1-m},\lambda_i>0$,forall$i=1,2,\dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,\dots ,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0\ge {\mu }_0$ on $\widehat{\mathrm{\Omega }}$ ($\widehat{R^n}$, respectively) for some constant ${\mu }_0>0$ and ${\gamma }_1>\frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $\widehat{\mathrm{\Omega }}$ (or $\widehat{R^n}$ respectively) to infinity as $t\to \mathrm{\infty }$. If $u_0\ge {\mu }_0$ on $\widehat{\mathrm{\Omega }}$ ($\widehat{R^n}$, respectively) for some constant ${\mu }_0>0$ and satisfies $\lambda_i|x-a_i|^{-\gamma_i}\le u_0(x)\le \lambda_i’|x-a_i|^{-\gamma_i’}$for$x\approx a_i$andsomeconstants$\frac{2}{1-m}<\gamma_i\le\gamma_i’<\frac{n-2}{m}$,$\lambda_i>0$,$\lambda_i’>0$,$i=1,2,\dots,i_0$,weprovethat$u$convergesin$C^2(K)$ for any compact subset $K$ of $\text{2}\mathrm{\Omega }\setminus \{a_1,a_2,\dots ,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to \mathrm{\infty }$.