Asymptotic behaviour of the finite blow-up points solutions of the fast diffusion equation
Shu-Yu Hsu · arXiv (Cornell University) · 2022
Let $n\ge 3$, $0\frac{n(1-m)}{2}$ which satisfies $λ_i|x-a_i|^{-γ_i}\le u_0(x)\le λ_i'|x-a_i|^{-γ_i'}\,\,\forall 00$, $λ_i'\geλ_i>0$ and $\frac{2}{1-m}0$, $u(x,0)=u_0(x)$ 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 $\partialΩ\times (0,\infty)$.