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

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in\mathbb{Z}^+$, $\Omega\subset\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$, $0\le f\in L^{\infty}(\partial\Omega)$ and $0\le u_0\in L_{loc}^p(\widehat{\Omega})$ for some constant $p>\frac{n(1-m)}{2}$ which satisfies $\lambda_i|x-a_i|^{-\gamma_i}\le u_0(x)\le \lambda_i'|x-a_i|^{-\gamma_i'}\,\,\forall 0<|x-a_i|<\delta$, $i=1,\dots, i_0$ where $\delta>0$, $\lambda_i'\ge\lambda_i>0$ and $\frac{2}{1-m}<\gamma_i\le\gamma_i'<\frac{n-2}{m}$ $\forall i=1,2,\dots, i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=\Delta u^m$ in $\widehat{\Omega}\times (0,\infty)$, $u(a_i,t)=\infty\,\,\forall i=1,\dots,i_0, t>0$, $u(x,0)=u_0(x)$ in $\widehat{\Omega}$ and $u=f$ on $\partial\Omega\times (0,\infty)$, as $t\to\infty$. 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 $t\to\infty$. We will also prove the existence of the minimal solution of $u_t=\Delta u^m$ in $\widehat{\Omega}\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\widehat{\Omega}$, $u(a_i,t)=\infty\quad\forall t>0, i=1,2\dots,i_0$ and $u=\infty$ on $\partial{\Omega}\times (0,\infty)$.