Asymptotic behaviour of blow-up 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, $a_1,a_2,\ldots ,a_{i_0}\in \Omega$, ${\widehat{\Omega }}=\Omega \setminus \{a_1,a_2,\ldots ,a_{i_0}\}$, $0\le f\in L^{\infty }(\partial \Omega )$ and $0\le u_0\in L_{loc}^p({\widehat{\Omega }})$ for some constant $p>\frac{n(1-m)}{2}$ which satisfies $\lambda _i|x-a_i|^{-\gamma _i}\le u_0(x)\le \lambda _i'|x-a_i|^{-\gamma _i'}\,\,\forall 0<|x-a_i|<\delta$, $i=1,\ldots , i_0$ where $\delta >0$, $\lambda _i'\ge \lambda _i>0$ and $\frac{2}{1-m}<\gamma _i\le \gamma _i'<\frac{n-2}{m}$ $\forall i=1,2,\ldots , i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution u of $u_t=\Delta u^m$ in ${\widehat{\Omega }}\times (0,\infty )$, $u(a_i,t)=\infty \,\,\forall i=1,\ldots ,i_0, t>0$, $u(x,0)=u_0(x)$ in ${\widehat{\Omega }}$ and $u=f$ on $\partial \Omega \times (0,\infty )$, as $t\rightarrow \infty$. 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 $t\rightarrow \infty$. 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 ${\widehat{\Omega }}$, $u(a_i,t)=\infty \quad \forall t>0, i=1,2\ldots ,i_0$ and $u=\infty$ on $\partial {\Omega }\times (0,\infty )$.