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, $a_1,a_2,\dots ,a_{i_0}\in \Omega$, $\hat{\Omega }=\Omega \setminus \{a_1,a_2,\dots ,a_{i_0}\}$, $0\le f\in L^{\infty }(\partial \Omega )$ and $0\le u_0\in L_{loc}^p(\hat{\Omega })$ for some constant $p>\frac{n(1-m)}{2}$ which satisfies ${\lambda }_i|x-a_i{|}^{-{\gamma }_i}\le u_0(x)\le {\lambda }_i^{\prime }|x-a_i{|}^{-{\gamma }_i^{\prime }}\forall 0<|x-a_i|<\delta$, $i=1,\dots ,i_0$ where $\delta >0$, ${\lambda }_i^{\prime }\ge {\lambda }_i>0$ and $\frac{2}{1-m}<{\gamma }_i\le {\gamma }_i^{\prime }<\frac{n-2}{m}$ $\forall i=1,2,\dots ,i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=\Delta u^m$ in $\hat{\Omega }\times (0,\infty )$, $u(a_i,t)=\infty \forall i=1,\dots ,i_0,t>0$, $u(x,0)=u_0(x)$ in $\hat{\Omega }$ 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 $\hat{\Omega }\times (0,\infty )$, $u(x,0)=u_0(x)$ in $\hat{\Omega }$, $u(a_i,t)=\infty \forall t>0,i=1,2\dots ,i_0$ and $u=\infty$ on $\partial \Omega \times (0,\infty )$.

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