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

Abstract

Let $\Omega\subset\R^n$ be a smooth bounded domain and let $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$ and $\widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=\Delta u^m$, $u>0$, in $\widehat{\Omega}\times (0,\infty)$ ($\widehat{R^n}\times (0,\infty)$ respectively) which satisfies $u(x,t)\to\infty$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots,i_0$, when $0<m<\frac{n-2}{n}$, $n\geq 3$, and the initial value satisfies $0\le u_0\in L^p_{loc}(\2{\Omega}\setminus\{a_1,\cdots,a_{i_0}\})$ ($u_0\in L^p_{loc}(\widehat{R^n})$ respectively) for some constant $p>\frac{n(1-m)}{2}$ and $u_0(x)\ge \lambda_i|x-a_i|^{-\gamma_i}$ for $x\approx a_i$ and some constants $\gamma_i>\frac{2}{1-m},\lambda_i>0$, for all $i=1,2,\dots,i_0$. 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 $\widehat{\Omega}$ ($\widehat{R^n}$, 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 $\widehat{\Omega}$ (or $\widehat{R^n}$ respectively) to infinity as $t\to\infty$. If $u_0\ge\mu_0$ on $\widehat{\Omega}$ ($\widehat{R^n}$, 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 $x\approx a_i$ and some constants $\frac{2}{1-m}<\gamma_i\le\gamma_i'<\frac{n-2}{m}$, $\lambda_i>0$, $\lambda_i'>0$, $i=1,2,\dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $\2{\Omega}\setminus\{a_1,a_2,\dots,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to\infty$.