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

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a smooth bounded domain and let $a_1,a_2,\dots ,a_{i_0}\in \Omega$, $\hat{\Omega }=\Omega \setminus \{a_1,a_2,\dots ,a_{i_0}\}$ 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 $u_t=\Delta u^m$, $u>0$, in $\hat{\Omega }\times (0,\infty )$ ($\widehat{R^n}\times (0,\infty )$ respectively) which satisfies $u(x,t)\to \infty$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots ,i_0$, when $0<m<\frac{n-2}{n}$, $n\ge 3$, and the initial value satisfies $0\le u_0\in L^p_{loc}(\2Ω\setminus\{a_1,\cdots,a_{i_0}\})$ ($u_0\in L_{loc}^p(\widehat{R^n})$ respectively) for some constant $p>\frac{n(1-m)}{2}$ and $u_0(x)\ge {\lambda }_i|x-a_i{|}^{-{\gamma }_i}$ 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 $\hat{\Omega }$ ($\widehat{R^n}$, 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 $\hat{\Omega }$ (or $\widehat{R^n}$ respectively) to infinity as $t\to \infty$. If $u_0\ge {\mu }_0$ on $\hat{\Omega }$ ($\widehat{R^n}$, respectively) for some constant ${\mu }_0>0$ and satisfies ${\lambda }_i|x-a_i{|}^{-{\gamma }_i}\le u_0(x)\le {\lambda }_i^{\prime }|x-a_i{|}^{-{\gamma }_i^{\prime }}$ for $x\approx a_i$ and some constants $\frac{2}{1-m}<{\gamma }_i\le {\gamma }_i^{\prime }<\frac{n-2}{m}$, ${\lambda }_i>0$, ${\lambda }_i^{\prime }>0$, $i=1,2,\dots ,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $\2Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to \infty$.

arXiv comment38 pages