论文标题
快速扩散方程的渐近溶液在穿刺的欧几里得空间中的渐近溶液
Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space
论文作者
论文摘要
对于$ n \ ge 3 $，$ 0 <m <\ frac {n-2} {n} $，$β<0 $和$α= \ frac {2β} {2β} {1-m} $，我们证明存在，独特性和差异性的生存，唯一性和渐进性，是奇妙的永恒自我相似的溶液的起源， $（\ MATHBB {r}^n \ setMinus \ {0 \}）\ times \ times \ times \ mathbb {r} $的形式$u_λ（x，x，x，x，t）= e^{ - αt}f_λ（e^{ - βT} x），x \ in \ in \ in \ mathbb {r} t \ in \ mathbb {r}，$其中$f_λ$是满足$$ \ frac {n-1} {m}Δf^m+αf+βx\ cdot \ cdot \ cdot \ cdot \ cdot \ cd = 0 $ \ underSet {\ ordack {r \ to 0}}} {\ lim} \ frac {r^2f（r） $ \ unterSet {\ orperack {r \ to \ infty}}} {\ lim} r^{\ frac {n-2} {m}} {m}} f（r）=λ^{\ frac {\ frac {2} {1-m} {1-m} {1-m} - \ frac {n-2} {n-2} {m}} $ noonstonants = 0 $ 0 $ noft and osand $ nost in oss oss osing。 结果，我们证明了快速扩散方程的库奇问题解决方案的存在和唯一性$ f_ {λ_1}（x）\ le u_0（x）\ le f_ {λ_2}（x）$，$ \ forall x \ in \ Mathbb {r}^n \ setMinus \ {0 \ {0 \} $ u_ {λ_2}（x，t）$，$ \ forall x \ in \ mathbb {r}^n \ setMinus \ {0 \}，t \ ge 0 $，对于某些常数$λ_1>λ_1>λ_2> 0 $。 我们还证明，当$ n = 3,4 $和$ \ frac {n-2} {n+2} {n+2} \ le m <\ le m <\ frac {n-2} {n-2} {n} $ holds时，我们还证明了快速扩散方程的这种单数解决方案$ u $ u $的渐近行为。 Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ is also obtained when $3\le n<8$, $1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)$,和$ u（x，t）$在\ Mathbb {r}^n \ setMinus \ {0 \} $中，在适当条件下的初始值$ u_0 $的适当条件下。
For $n\ge 3$, $0<m<\frac{n-2}{n}$, $β<0$ and $α=\frac{2β}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in $(\mathbb{R}^n\setminus\{0\})\times \mathbb{R}$ of the form $U_λ(x,t)=e^{-αt}f_λ(e^{-βt}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R},$ where $f_λ$ is a radially symmetric function satisfying $$\frac{n-1}{m}Δf^m+αf+βx\cdot\nabla f=0 \text{ in }\mathbb{R}^n\setminus\{0\},$$ with $\underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}}=\frac{2(n-1)(n-2-nm)}{|β|(1-m)}$ and $\underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r)=λ^{\frac{2}{1-m}-\frac{n-2}{m}}$, for some constant $λ>0$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=\frac{n-1}{m}Δu^m$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$ with initial value $u_0$ satisfying $f_{λ_1}(x)\le u_0(x)\le f_{λ_2}(x)$, $\forall x\in\mathbb{R}^n\setminus\{0\}$, which satisfies $U_{λ_1}(x,t)\le u(x,t)\le U_{λ_2}(x,t)$, $\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0$, for some constants $λ_1>λ_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ is also obtained when $3\le n<8$, $1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)$, and $u(x,t)$ is radially symmetric in $x\in\mathbb{R}^n\setminus\{0\}$ for any $t>0$ under appropriate conditions on the initial value $u_0$.
