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

Abstract

Let $n\ge 3$, $0 \frac{n(1-m)}{2}$ which satisfies $λ_i|x-a_i|^{-γ_i}\le u_0(x)\le λ_i'|x-a_i|^{-γ_i'}\,\,\forall 0<|x-a_i|<δ$, $i=1,\dots, i_0$ where $δ>0$, $λ_i'\geλ_i>0$ and $\frac{2}{1-m}<γ_i\leγ_i'<\frac{n-2}{m}$ $\forall i=1,2,\dots, i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=Δu^m$ in $\widehatΩ\times (0,\infty)$, $u(a_i,t)=\infty\,\,\forall i=1,\dots,i_0, t>0$, $u(x,0)=u_0(x)$ in $\widehatΩ$ and $u=f$ on $\partialΩ\times (0,\infty)$, 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 $u_t=Δu^m$ in $\widehatΩ\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\widehatΩ$, $u(a_i,t)=\infty\quad\forall t>0, i=1,2\dots,i_0$ and $u=\infty$ on $\partialΩ\times (0,\infty)$.