论文标题
快速扩散方程的有限爆炸点解决方案的渐近行为
Asymptotic behaviour of the finite blow-up points solutions of the fast diffusion equation
论文作者
论文摘要
令$ n \ ge 3 $，$ 0 <m <\ frac {n-2} {n} $，$ i_0 \ in \ in \ mathbb {z}^+$，$ω\ subset \ subset \ mathbb {r}^n $ be be a _1，a_1，a_2，a_2，a_2，a_2，a_2，y_2，$ ______________________ $ \wideHatΩ=ω\ setMinus \ {a_1，a_2，\ dots，a_ {i_0} \} $，$ 0 \ le f \ in L^{\ infty}（\partialΩ）in l^{\partialΩ）和$ 0 $ p> \ frac {n（1-m）} {2} $满足$λ_i| x-a_i |^{ - γ_I} \ le u_0（x）\leλ_i'| x-a_i'| x-a_i |^{ - γ_i'} i_0 $其中$δ> 0 $，$λ_i'\geλ_i> 0 $和$ \ frac {2} {1-m} <glac \ lecy \leγ_i'<\ frac {n-2} {n-2} {m} {m} $ $ $ \ $ \ forall i = 1,2，\ dots，\ dots，\ dots，i_0 $ $ $ comantants。我们将证明有限的爆炸点解决方案的渐近行为$ u _ $ u _ $ u_t =Δu^m $ in $ \wideHatΩ $ \wideHatΩ$和$ u = f $ $ \partialΩ\ times（0，\ infty）$，as $ t \ to \ infty $。我们将在有限的圆柱域中构建有限的爆炸点解决方案，具有适当的横向边界值，以使有限的爆炸点解在两个给定的谐波函数之间振荡为$ t \ to \ to \ infty $。我们还将证明存在$ u_t =δu^m $的最小解决方案，in $ \ \wideHatΩ $ u = \ infty $ on $ \ \partialΩ\ times（0，\ infty）$。
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in\mathbb{Z}^+$, $Ω\subset\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\dots,a_{i_0}\inΩ$, $\widehatΩ=Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$, $0\le f\in L^{\infty}(\partialΩ)$ and $0\le u_0\in L_{loc}^p(\widehatΩ)$ for some constant $p>\frac{n(1-m)}{2}$ 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 $\partialΩ\times (0,\infty)$.
